Languages are more than just collections of words, and you're going to run into many problems at many levels.
Let's pick one really obvious problem: What counts as a word? The single Yupik word "tuntussuqatarniksaitengqiggtuq" crams an entire clause into it. The root "ssur" just means "to hunt", but after attaching 6 affixes, the full word means "He had not yet said again that he was going to hunt reindeer." Surely you don't want every possible Yupik clause to be a word in your language. But if you want each Yupik lexeme to be a word, then how do they incorporate, or attach to, non-Yupik lexemes that don't follow the same rules?
And even the niggling details will be problems. Do adjectives go before nouns (like English) or after (like Hebrew)? As long as you pick one, that's fine; you can handle all adjectives and nouns from English, Hebrew, Japanese, Persian… but what about Romance adjectives? Some come first, some go second, and some can go on either side and have a different meaning each way. How do you get both the pre-noun and post-noun meanings of "grande" in your language? But you can't just use the Romance system, because that would make phrase boundaries too ambiguous to parse, were it not for the extremely restricted set of word endings in each Romance language (which will, of course, not be true for your language).
But I think there's a more tractable version of your problem that might still be interesting (depending on what interested you in the question).
Real languages borrow words all the time, but they don't do it exactly. Look at French "tenez" (you-plural hold!) to English "tennis" (the familiar racket game) to Japanese "tenisu" (the same game). You can see that the phonological, semantic, and morphological changes are sometimes tiny, sometimes huge ("tenisu" and "tennis" are almost perfect synonyms, but "tennis" and "tenez" aren't even the same part of speech, or about the same kinds of things).
And all of this works just fine. The borrowed word takes on a life of it own in its new language. So, if you could simulate realistic borrowing, and then just feed in every word from every language…
But can you simulate realistic borrowing? Depends on how realistic. The rules for how borrowing works are partly regular (e.g., the same English syllables usually map to the same Japanese syllable sequences), but only partly. More seriously, most of those rules are well below conscious accessibility. And most of them involve both the local context (why someone felt the need to borrow that word), and language-internal context (what existing words it has to fit into its new lexicon with). But you don't need to exactly model what would happen if a real community of X-speakers felt the need to borrow word Y from language Z, just to come up with an algorithm (probably randomizing a lot of the details) that feels realistic, roughly, and (depending on your intended goal) that may be more than interesting enough.
A few notes:
- You still have to solve the "what is a word" problem, but now you can solve it with just "morphemes are words".
- Given that the existing lexicon you're borrowing into is part of what affects borrowing, you probably want to iterate over all of the words of all of the languages in random order.
- Function words and morphemes like "the" and past-tense "-d" are almost certainly going to end up distorted to the point of unrecognizability by any borrowing algorithm—maybe that's acceptable, or even fun, or maybe you just skip them.
- Almost every part of the core design of your original language will affect how much things get distorted. (Think of the way Japanese has to insert "u" into closed syllables, import most adjectives as nouns, etc.) But this is something you can—and probably want to—play with, not a problem to surmount.
What I think really makes it infeasible in practice is just sourcing the lexicons of all of the world's languages. You can't just "use a dictionary", unless you work out how to parse dictionary-style definitions (in every language in the world, and mostly written in an unusual style that would confuse general-purpose parsers like the ones Google has trained up…) into the kind of representation you need (look at, e.g., an HPSG lexical entry, for something closer to usable by a plausible word-borrower algorithm).
But fortunately, you don't need to do 100% of the world's words to have something interesting. Just manually creating entries for a few hundred words from a dozen languages is probably already enough to start playing with your algorithm. If that's then so promising/exciting/whatever that it's worth embarking on a decade-long dictionary-parsing project, great; if not, go as far as you want to go with it.