We have a 401(k) with employer matching that begins after 12 months of service and also 100% vesting that begins after 12 months of service. Employees can contribute after 3 months.

Lots of employees come and go. If an employee works one year or so, leaves, then returns, can we resume the employer matching and 100% vesting after a shorter time period such as 3 months?

You can delete from the plan "break in service" provisions that ignore prior service of nonvested participants.