I don't know you or your operation so take this advice for what it's worth. I was raised on the farm, left the farm to work in 5 different businesses, then came back to the farm and am now the owner. I've been in both sides of the fence.  I have observed over the years if the boss doesn't do it, nobody does it.

If the boss keeps his paperwork caught up, his tools clean and organized, and his vehicle maintained ... it doesn't mean everyone does.  But if the boss slacks off on any of these I guarantee the employees will follow his lead.

The other part of that is responsibility.  I hate laying blame.  My operating style is "OK, we have a problem let's fix the problem.  I don't care who did it, let's fix it."  But part of being management is from time to time placing blame where it belongs.  Not to be vindictive about it but to identify problems and solve them.  If the trucks are to be maintained.  When it isn't done find out who didn't do it and explain it WILL be done this way.  If it is all 5 employees, all 5 need to understand it all be done. If they are not willing to do what is required they are free to work elsewhere with different requirements.

These dynamics change when you are dealing with other owners or family members,  but the basics still apply.   If they are not willing to do what needs done they should do it elsewhere.

Having said all that, let's go back to where I started. If the boss isn't doing it nobody will do it.  That includes making or allowing time for it.  One of my employers always over-booked the schedule.  We finally learned if your truck needed serviced you put servicing your truck on the schedule and then made sure you did it.

So what you have to figure out is are they not doing this because they don't have time, because it is not planned, or just because they don't want to and the boss lets them get away with it?

edit:  Oh, and having worked where nobody had an assigned vehicle and where everyone had one assigned to them, I recommend if at all possible assigning responsibility for a vehicle to a person.  He may not be the one using it all the time (for instance we had one service truck with a crane for tank setting) but he is responsible for it. When Charlie's truck came back and the tools were not in the right place and it was not restocked the last user knew about it.  Even (especially) if it was the boss who drove it last.