Coffee options for those with a weak stomach? My stomach doesn't take coffee kindly for some reason, but I do like the taste and smell of it.

Are there any coffee types out there for those with a similar issue?
Preparation method will have a big factor in extraction, acidity (such as in @Patrick's answer) among them.  The method used (temperature, time, technique, filtering method) will have a big impact on what comes out in the cup. Astringency and bitterness might also be factors that have physical effects for you; see this article for a discussion of the differences between the two, causes, and related factors.

Try cold brew. Cold brew coffee will have lower acidity, and might be a good option for you; here's an article that discusses different preparation methods, but in the cold-brew section it has an anecdote of someone with similar "stomach".  Here's another on cold-brew and acidity. See also the cold-brew tag.

Paper filtering seems to remove some acids, but it's not clear whether these are the type that might argue with your stomach. For example, The AeroPress FAQ claims that their paper filter creates lower acidity brew, but I can't immediately find a better, proper reference.

Try some antacid. If you're experiencing heartburn, try a calcium-based antacid before and/or after drinking the coffee, and see if that changes things. (Disclaim: This isn't medical advice, don't take it if you shouldn't, talk to a doctor about meds, etc.).

Science! If you're into thicker scientific reading, here's a paper on extraction methods and outcomes, if you're into that sort of thing. It discusses various aspects of acids and acidity. It looks pretty dense.