Yes you can. Back in the day I was a TA Medic (Shit, my dirty little secret's out) I got my lightweights, KF shirt and JHW covered in someone else's blood after he suffered a huge gash to his thigh by sliding up the seats in the back of a Bedford that stopped suddenly. When we returned to the TAC from the hospital the QM told me to hand in my uniform for laundering so that it didn't contaminate my own machine. I got it back a couple of weeks later (un-ironed).
There is a laundry service for most if not all units, however most units I've been at provide washing/drying facilities on camp as it is cheaper/less admin that the laundry system. Issue Dhobi dust is normally available through the system too. In NI our washers and driers were shared with the TA lads and lasses.