I don't know if this is necessarily a question with a specific answer that can be backed up by historical evidence. Today's "drink" has been the standard spelling for several hundred years, and can be traced back to Old English drinc (noun, the form here) and drincan (the verb form), both of which ended with the same sound we have today.

The simplest explanation is that it's just a mistake, similar to somebody making a typo. For obvious reasons it's hard to back that up with hard evidence, but I struggle to look past that unless there's some evidence that "dring" was in widespread use beyond this particular cart. It's easy to overlook an error of just one letter, particularly when the sound suggested by the <g> is very similar to the correct <k> (they differ only in voicing*). Especially if the sign-maker and/or cart-owner are non-native speakers of English, which (and I must say that here my interest is in linguistics, not historical demographics, so I may be wrong on this point - somebody please correct me if so) I don't think is especially unlikely in 1940s New York, this could easily sneak past people unnoticed.

* in fact, <ng> in English generally implies /ŋ/, while <nk> would imply /ŋk/, so in fact the difference here isn't just the voicing if you analyse the sounds actually produced by a native English speaker. But a typical speaker (i.e. a layman with no experience in linguistics) would likely be unaware of this, and the difference between the likely pronunciation of 'dring' vs. 'drink' is in any case small enough that I believe this argument holds.

I realise this post is rather light on hard evidence for a top-level post in this sub, but I hope it's acceptable as I think that's basically unavoidable if my answer is correct. The etymological origin of 'drink' is the only thing I said which I think can be reasonably sourced, personally I referred to the Concise Oxford Dictionary of English Etymology, though it should be verifiable from any English etymological dictionary.

Also, anecdotally, I can provide an image of a chip shop near my parents' house which has a sign offering "bergers". I'm not trying to use it as evidence that my suggestion must be correct for this particular case, it just came to mind as an example of something similar. I'll be happy to remove this if the mods think it's inappropriate.

Where was that picture taken?