You could modify word2vec to embed word fragments as part of the training process. You could also use stemming before training, or if you have a decent amount of computational resources you could embed trie entries with word2vec representations of word fragments and probabilistic models of the likely next character/syllable/word, which would allow you to use something like a markov process

For word completion, I would create a trie and encode naive char-to-char and absolute (word prefix vs. all recorded) frequencies for each edge during its creation, noting that the "end word" value is also possible and deserves a frequency. Then when a user is entering a single word without prior context, you simulate a markov process from the last character to the end of the word. If the user has input a short but unlikely combination you can observe the frequencies of untraversed edges from the nodes of the current path and start a markov process from there if it is much more likely. That gets you to the end of your current word in terms of its string representation. From there you can use n-grams if desired, or go straight into sanitization preceding vectorizaton, to construct a likely query

If I were you I would decouple processes in your pipeline. I mentioned the best way I know of combining word vectors and word fragments (embedding word fragments of a corpus into word2vec, then indexing them with a trie) and I don't think it would be feasible for size reasons - although perhaps the topic of this thread could make it more computationally amenable. It sounds like what you want to do is 1 first infer a word/phrase, 2 stem/sanitize it, 3 map it to its word2vec representation, then 4 do some search query using the vector. 2 and 3 could be combined if desired (would decrease corpus vocab substantially and be good space-wise, improve embeddings of stems of rare words by reducing overfitting, but lose some semantic complexity), perhaps even aggressively, but not with 1 unless you further modify the training process / augment the corpus with fragments.

TL;DR: There's not a good way I know of to use a word2vec mapping trained on a vanilla corpus to directly account for short spelling errors since individual spelling error fragments will be rare or not present within the vanilla data. You seem to think Levenshtein will help but keep in mind this is an expensive pairwise string comparison algorithm. Unless you implement a good way to check which strings to compare the input fragment to, you will likely perform too many comparisons because you won't know where to start