As quick as we can?

The article Flat Adverbs Are Flat-Out Useful suggests that "quick" is a flat adverb.

Just as an example, "come quick" and "come quickly" are both grammatically correct.

Consider the similar sentence: "We'll get back to you as soon as we can." Here, "soon" is taking the place of "quick" with all most the exact same meaning ("fast" works as well).

Consider an alternative sentence: "We'll do it as quick as we can." and a sentence with similar structure but different meaning: "We'll do it as quietly as we can." Here the word "quiet" is almost never presented as a flat adverb (maybe never?) and so you need the -ly.

It's interesting to note that AHDEL lists the adverbial usage as synonymous with quickly, and with no caveats.

Collins lists the adverbial usage as being in an informal register.

I'd certainly choose 'quickly' for Michael's delightfully highbrow example, but would use either variant in OP's example.

I'd say it was acceptable, since it's a common usage. It's not quite correct, however. "Quickly" is the proper adverb.