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
Nom. eḥ | ayau | ayaḥ
Voc. eḥ/e | ayau | ayaḥ
Acc. am | ayau | ayaḥ
Ins. ayā | ebhyām | ebhiḥ
Dat. aye | ebhyām | ebhyaḥ
Abl. eḥ | ebhyām | ebhyaḥ
Gen. eḥ | ayoḥ | ayām
Loc. ayi | ayoḥ | eṣu
Monier-williams Dictionary also gives the word ve - a bird and se - service (f.) and serving (n.). It is declined like above.