Can umpires start a replay review without a manager challenge?

Yes, in two situations:

• At any time in a game, the crew chief may start a review of a potential home run call without a manager challenge being used.
• Beginning in the eighth inning, the crew chief may review all reviewable calls on his own initiative, or at the request of a manager who has no manager challenges left.

Source: MLB, Replay Review (mlb.com/glossary/rules/replay-review).