There are no natlangs I'm aware of that have been conclusively shown to entirely lack a lexical distinction between nouns and verbs although some come close (e.g. Salishan languages).
However, it is possible among conlangs. Lojban lacks such a distinction, although its Brivla are much more verb-like and you're asking about languages with more noun-like classes of content words.
Most of my conlangs have a class of content words which is easier to regard as more noun-like because they are unmarked as subjects and marked as predicates. (I have a couple with more verb-like classes, unmarked as predicates and marked as subjects.) To be very clear, I am only claiming the lack of these as separate, identifiable lexical classes, not claiming that the sentences lack subjects or predicates.
I'll show some examples as glosses. The general sentence structure is a subject consisting of an unmarked content word phrase (which I'll abbreviate as NP for convenience), which is followed by a predicate, consisting of a NP which is introduced by an invariant predicate marker. (This predicate marker could be argued to be a verb, a copula, meaning that there is a closed class of verbs consisting of only one member.) The equivalent of objects are marked as possessors (introduced by a genitive preposition "GEN", which can be dropped in a lot of cases and also has a number of contracted forms with certain lexems, but I'll always gloss it separately here for maximum clarity).
husband GEN 1S PRED hunter GEN bird
"My husband hunts birds."
"My husband is a bird hunter."
Any content word can stand as the head of the subject or the head of the predicate without needing to be marked. A sentence can be flipped about the predicate marker without any unexpected changes in meaning.
Any irregular changes in meaning would point towards zero-derivation rather than a lack of a lexical class distinction, for example "the fish jumps" vs "*the jump fishes". In cases like this, differing meanings between "the fish" and "to fish" and "to jump" vs "a jump" mean that the nominal meaning and the verbal meaning are separate (though related) lexemes, not simply one lexeme that can be used freely in any syntactic position.
In the language I'm glossing here, subject position implies definiteness and predicate position may imply indefiniteness. Semantically, however, some noun phrases of course have default definite or indefinite interpretations. In a monogamous society, "husband GEN 1S" is likely to be interpreted as the definite "my husband" rather than as the indefinite "a husband of mine", even in the predicate.
hunter GEN bird PRED husband GEN 1S
"The hunter of birds is my husband."
"The one who hunts birds is my husband."
Translating ditransitive sentences as well as any other sentences involving adjuncts requires the use of what I call a situational clause, which is a kind of subordinate clause that can modify a nominal phrase (in either the subject or the predicate). They are introduced by a particle I gloss as SIT. A handy shorthand to think of the meaning of "SIT" is "whereby" or "with [subject] being [predicate]".
husband GEN 1S PRED giver GEN bird SIT recipient PRED 1S
"My husband gives me a bird."
"My husband is a giver of a bird whereby the recipient is me."
giver GEN bird SIT recipient PRED 1S PRED sleeper SIT location PRED bed GEN 1S
"The one who gives me the bird is sleeping in my bed."
CONT communicator SIT hearer PRED sister GEN 2S PRED who
"Who is the person who is talking to your sister right now?"
1S PRED PST wash GEN floor SIT benefactor PRED 2S
"I mopped the floor for you."
I could of course also indicate subjects and predicates within the subordinate clauses too as below, but it's pretty unnecessary from this point on.
1S PRED late.entity SIT cause PRED COMP 1S PRED NEG hearer GEN alarm
Matrix clause: [SBJ] [PREDICATE_________________________________________________________]
SIT clause: | [SBJ] [PRED___________________________________]
COMP clause: | [SBJ] [PREDICATE______________]
"I'm late because I didn't hear my alarm."
"I'm late, with the cause being that I didn't hear my alarm."
The examples above mostly don't indicate optionally marked grammatical categories such as TAM and definiteness. These grammatical categories can be applied to all content words and thus cannot be used to divide the lexicon into nouns and verbs.
PST husband GEN 1S
"my ex husband"
PST teacher GEN 1S
"my former teacher"
PST giver GEN bird
"one who gave a bird"
"one who was tired"
Definiteness rarely needs to be indicated in the subject as that position implies definiteness, but is contrastive in the predicate.
1S PRED teacher
"I am a teacher."
1S PRED DEF teacher
"I am the teacher."
"I'm the one who teaches."
1S PRED PST giver GEN bird
"I gave a bird"
"I'm one who gave the/a bird."
1S PRED DEF PST giver GEN bird
"I'm the one who gave the/a bird."
(This can often be achieved by reversing the sentence, however: PST giver GEN bird PRED 1s "The one who gave the/a bird is me."/"It's me who gave the/a bird.")
A structure using the content word "situation" is analogous to a gerund or infinitive and this, too, is not restricted to a particular subset of content words which we could identify as a separate class.
situation teacher PRED good.entity
"Teaching is good."
"Being a teacher is good."
"It's good to teach."
"It's good to be a teacher."
situation 1S PRED difficult.entity
"It's hard to be me."
"Being me is difficult."
situation giver PRED cause GEN COMP (giver) PRED happy.entity
"Giving makes you happy."/"Giving makes the giver happy."
Modal-verb-like meanings can be used in conjunction with any content word, regardless of its semantic type.
1S PRED wanter teacher
"I want to be a teacher."
"I want to teach."
1S PRED wanter PST husband GEN 2S
"I want to be your ex husband."
1S PRED wanter bird
"I want to be a bird."
1S PRED wanter (possessor) GEN bird
"I want to have a bird."
1S PRED able.entity teacher
"I can teach."
"I can be a teacher."
able.entity PRED actor but NEG able.entity PRED teacher
"Those who can, do; those who can't, teach."
Essentially, everything that could be thought of as a verb is represented by an agent noun (which is its own morpheme, not derived from a non-existent verb). At this point, you might point out that there is a small difference between teaching and being a teacher, and this is generally either simply aktionsart ("I am a teacher" implies a much longer time span than "I am teaching"), which is easily handled by TAM marking as necessary, or some other connotation (e.g. "I am a teacher" implies that it is a job, whereas "I teach" does not necessarily), which can be handled by additional words.
1S CONT teacher
"I am teaching."
1S HAB teacher
"I am a teacher."
1S worker teacher
"I am a (professional) teacher."
"I teach (as a job)."
I've been making conlangs like this for probably around 15 years now and, once I got into the swing of it, haven't encountered any issues or ambiguities when translating things. The few people who have tried to use my conlangs have often made errors that indicate that they expect the flexibility to be in the form of zero derivation, such as expecting words with semantically verb-like properties ("actiony words") to function as gerunds in nouny-syntactic positions like subject or genitive modifier. The fact that there are no natlangs that are conclusively proven to lack a noun/verb distinction entirely may come down to a human cognitive bias and it may be that a language like mine, if spoken by a community of speakers over a long period of time, may spontaneously begin to divide up the lexicon. On the otherhand, a lexical distinction into nouns and verbs may simply be part of the "Earth sprachbund" and people expect to find it simply because it is in the languages they know. That's just conjecture in any case. My point is that it's possible for conlangs to lack such a lexical distinction. My own conlangs haven't been heavily tested, but Lojban has and the lack of a distinction between lexical classes of nouns and verbs is not one of its issues.