I did look it up. I also looked up "cafe", which came back with "café". However, "cafe" is at least as common in modern usage. All of the citations it gives are for "café" apart from one, which sneers at the accent-less version. That one, however, is from 1929, so can't be taken as being authoritative for current usage.

Now, if the OED is obviously ambiguous (and maybe out of date) for a relatively common word like "cafe", then it's reasonable to think that it may be likewise for a rare word like "jalapeno". See also the OED entry for "relais", where they give the definition as "in France: a restaurant or cafe (sometimes also providing overnight accommodation)". Note the lack of accent.

As for "Don’t go running to the OED for help in writing English poorly: they actually know how to spell", you have two problems here. First, you are being rude. Second, you misunderstand the point of the OED, which is to record usage, not to prescribe usage. Or perhaps you're accusing them of poor spelling in the entry for "relais", in which case you are therefore obliged to not cite the OED when accusing me of not being able to spell.