Yes, this might be a scam.

If the person wants their money via the internet (Green dot card, etc), that would be highly suspicious.
Suggest you tell them to come to the restaurant to pick up a check, and sign a release.  Tell them to bring a copy of their card invoice (if they paid by card).  Also tell them you will be taking their photograph.

I doubt if they will call again.
But, if they DO arrive at your restaurant, you really should have them sign a release, that includes some of their personally identifying information, such as home address and a driver's license number.  Also, if they claim they paid by card, but do not have the invoice, tell them you will check your records to see if that card number was actually used on the date they claim.

If somebody =really= got sick from eating your food, they would not beg for a small refund ... they would at least file a claim in small claims court, for medical treatment and "pain and suffering"!  Your assumption that this is a scam is probably valid.

You might want to contact local law enforcement, this sounds like a extortion scam.

Hi thank you for your comment.  He just called again this morning. We are being harassed by someone named "Michael Garcia."  He's saying he got sick from our food when our food is always cooked properly. He's asking for $30.  We asked  for his hospital bills or receipt if he can provide one so we can refund him, but he got angry and just wanted a refund.  He's a scam.  Yes, beware all restaurant owners.

Beware restaurant owner in Sacramento.
I have a small Japanese restaurant. Last week someone called this number said his girlfriend got sick from my sushi and he wants refund. I told him my fish always fresh, and always keep them proper temperature. He said your sushi always good but last week his girlfriend got sick. I'm in business in 2 years, never happened this before, but I said him he can get refund and he drove by get money 23$. But I noticed that I never seen him before. Today I got call from the same number saying he ate my restaurant last week his girlfriend got sick from my sushi.  I instantly recognized his voice. I said him I scammed me month ago, so you and your girlfriend should come to my restaurant with ID. He got angry and he said he called last Friday and wants his money back. I always here in my restaurant and never got call from him last Friday. I think he got confused  my restaurant number with different restaurant to scam. He threatened me he gonna call authority to report. I told him what he is doing wrong and he cannot rob people. If he is really hungry and don't have money, I would give something to eat. But robbing people this is not way young people live.