"to hold your alcohol" means to drink a reasonable amount of alcohol without feeling drunk. Some people are genetically unable to tolerate much alcohol. Some others drink so little that their bodies are not used to consuming larger quantities which would be reasonable for many people. Such people cannot "hold their drink / alcohol".

I will try to explain it. Think of holding = ie. 持つ. When you drink, your stomach "holds" the alcohol. If someone does not have a tolerance for alcohol...吐くかもしれません. Thus, their stomach would not "hold" the alcohol anymore. Does this make sense? If not, I can try to explain more^^