Normally we deal with environmental microbes without major problems.
Missing an arm, things can be different though.
First of all, there is just one hand to do everything with. If you go to the bathroom, the same hand cleans butt, shakes off drops, closes zipper and button, and handles door handles. Being one handed accumulates microbes for sure. It complicates matters in that you either end up building a lot more resistance or wash your hand a whole lot more often.
Secondly, the stump can get really sensitive to microbes. When I wear the prosthetic arm, the skin is soggy all day, then it risks drying out faster, and so even smaller amounts of otherwise harmless microbes can be a problem. That is why I do not like warm-up exercises on the floors surrounding swimming pools. Also I don't like to use warm tap water at home to wash my stump.
Thirdly, if you have an open wound, after an injury or if you get surgical stump osseointegration implants, you may seriously have to watch your back. You should probably stay away from public toilets, and definitely stay away from public swimming pools. Fourthly, I'd be extremely cautious with an organ or hand transplant. Bacteria and fungi are a simple reality of public life. Either you take a risk, or stay out of harm's way.
Just to give you an idea about the extent of the problem, here is an overview.
(Source: Flores et al., 2011 / [link])