A football player, momentarily at rest on the field, catches a football as he is tackled by a running player on the other team. This is certainly a collision (inelastic!) and momentum must be conserved. In the reference frame of the football field, there is momentum before the collision but there seems to be none after the collision. Is linear momentum really conserved? If so, explain how. If not, explain why.

Yes, the linear momentum of the system really conserved. When the running player strikes the player who is at rest, the two players will not come to rest suddenly, they move some distance and then they will come to rest. After collision the momentum of the football and running player will transfer to the momentum of rest player. So the two players will move with some distance and then they will come to rest. Thus the linear momentum of system before collision will transfer to the linear momentum of the system after collision. Therefore the linear momentum of the system will be conserved.