Yes, Sanskrit has stems which end in e. Examples are the sup pratyaya (nominal suffix)
ṅe or the tiṅ pratyayas (verbal suffixs)
e, se, te, āte, ante, dhve. These pratyayas (suffixs) when used as nouns can take all the vibhaktis (declensions). You would find them being used in sanskrit commentaries and grammar texts.
Ex: the Pāṇini sūtra 7.1.13:
ṅer yaḥ - here
ṅe is treated as a noun ending in e-stem, and declined in sixth/genitive case.
Another example is Harināmāmṛta-vyākaraṇa sūtra 162:
kṛṣṇāt ṅer yaḥ
After a masculine word ending in a-stem (called Kṛṣṇa in that grammar)
ṅe (the dative singular suffix) is replaced with
ya. One finally gets
kṛṣṇāya from this.
Another example is the single letter e which means Lord Viṣṇu according to Ekākṣara-kośa, which is declined as follows:
Mas. Sing. Dual Plural
eḥ | ayau | ayaḥ
eḥ/e | ayau | ayaḥ
am | ayau | ayaḥ
ayā | ebhyām | ebhiḥ
aye | ebhyām | ebhyaḥ
eḥ | ebhyām | ebhyaḥ
eḥ | ayoḥ | ayām
ayi | ayoḥ | eṣu
Monier-williams Dictionary also gives the word
ve - a bird. It is declined like above.