Submitting to a journal that requires a French abstract as a non-French-speaker: how important is the French abstract? I'm a non-French-speaking mathematician submitting an English paper to a French journal that requires a French abstract (but publishes papers in English).  I don't think I know anyone I'd be comfortable asking for help.

I took at stab at a translation using a combination of Google Translate and phrases found actual French papers.  I think my attempt is probably not so far from the mark, but would be no doubt cringe-inducing for an actual Frenchman.

Should I submit with no abstract and hope for the best, submit with the bad one, or is it important to find somebody to help me with a proper translation?
I would not use Google Translate for this. I recently moved to France and I needed to do this a few times, and when I did a Google Translate (even a 'nice' one*) it still sounded bad to the native French people around.  In that particular experience, a French person editing my 3 page document took about 1.5 hours. GT leaves much to be desired. Even when things are technically correct, it will be obviously non-French, which can have a bad effect on the initial reception of the document. 

One would think this isn't relevant to the scientific content, but in France, things like this are considered to be culturally important (which is why an English language journal requires a French abstract - PhD theses have the same requirement). I think your paper would have a better reception with a linguistically correct abstract, both from the people considering the submission and the people eventually reading and citing it.

There are services online which offer translations for a small sum of money. I'm not sure if I can link to some without it being considered spam, but they're easy to find. So based on my experience, this is what I recommend. They charge per-word so an abstract should be fairly inexpensive.

Like the commenter guifa suggested, speaking with your foreign language department would be another way to accomplish this without having to pay a translator. 

*If you HAVE to use Google Translate (which I don't recommend), I would translate the result back into English to make sure that words haven't been misinterpreted. I would also check with a good online French-English dictionary. My chosen one is Linguee. This is the most effective if you write your sentences very simplistically and concisely.