I'll check we have this medicine in stock

No, it isn't strictly correct for formal/written english, but its a reasonably common contraction in general use.

Including if would be far more common, and whether too, but the latter is a little more formal - more likely to be written than spoken.

As mentioned in the comments, this construction is a shorter version of "I'll check that we have this medicine in stock". The words "if" or "whether" cannot regularly be dropped in this kind of context, but the word "that" can be.

"I'll check we have this medicine in stock" doesn't sound great to me, but I wouldn't say that involves contraction or elision of either "if" or "whether".

Collins provides the following example for the use of "check" with a that-clause:

He checked that he had his room key.