Depends.
Is your definition of "crime" exclusively related to the crimes system? If yes, then you can absolutely go ahead and play the game without ever doing crimes. You'll lose an avenue of income (crimes and OCs) and a tonne of merits but you can still do other things with your time.
Does it also include actions related to energy like attacking, mugging and hospitalizing people? If yes, then it's gonna be hard. You can just train your energy, sure, but few factions will be willing to take you in if you refuse to attack people and some events like the upcoming halloweek will be unavailable to you. On the other hand, there's still reviving (effectively, being a doctor) and reviving factions where you can fit in just nicely (ask DocFox about it, he's a pacifist and a reviver, and he's doing just fine).
Then there's flying and companies, both completely unrelated to doing crimes (for the most part...).
There's also other activities like racing, hunting, stock trading and just trading in general, none of which are criminal activities.
More than anything, you'll be hurting for merits since most merits come from attacking and crimes.
TL;DR: You can but you'll close yourself off from a few sources of income. That said, there's plenty of other things you can do instead.