Q:

Can stagnate be used as a noun? 

I couldn't remember stalemate for some reason so opted to use the phrase, "put us in a stagnate".  In this context, would that be appropriate?  
Full sentence: I'm worried that a lack of coordination would put us in a stagnate. 
None of the dictionaries I've seen seem to approve of it, but I can't help but feel this is a fine usage of the word. 

A:

If we're talking about commonly recognized usage, then stagnation is the noun and stagnate is the verb.  Here is stagnation, a noun, in Merriam-Webster:

: a stagnant state or condition : a state or condition marked by lack of flow, movement, or development

And here is stagnate, a verb:

: to become or remain stagnant

So for your example sentence, it would be better to write:

I'm worried that a lack of coordination would put us in stagnation. ( using the noun form)
I'm worried that we would stagnate if we do not coordinate. ( using the verb form)

It's hard to say why stagnate would sound fine as a nonce noun to you.  For example, the -ate suffix marks stagnate as a verb from Latin, and several other verbs share that form (segregate, aggregate, aggravate, ingratiate) (OED, "-ate, suffix3").  That said, there are also nouns that have an -ate ending (senate, prelate, mandate) (OED, "-ate, suffix1").  So it would be possible to hear the word with its ending and think, "This sounds noun-ish as well as verb-ish." 
That said, this hypothetical usage is rare enough that the Corpus of Contemporary American English turns up zero results for "stagnate_n" (stagnate as a noun) and "a stagnate" (stagnate with an article, so either a noun or adjective), but it turns up 152 results for "stagnate_v" (stagnate as a verb).  Their largest corpus, NOW: News on the Web, turns up zero results for "stagnate_n" but 3857 results for "stagnate_v."  Strangely, "a stagnate" does turn up ten results, but they all use stagnate as an adjective in a peculiar fashion.  It feels like a misspelling of stagnant.  A few examples:

it's a stagnate case
they just look like a stagnate team, going through the motions. 
including a stagnate and aging population

So if stagnate is ever used as a noun, it's so rare that a misspelling of the adjective stagnant is more common.