The Log Reader agent does not apply transactions to the subscriber database. Its role is to extract transactions from the transaction log of the published database and store them in the distribution database.
The distribution agent is responsible for reading relevant transactions from the distribution database and appying them to the subsciber database. A distribution agent can be either push or pull (pull subscription agent).
Both of these agents operate independently of each other. You can stop either of them without having a negative impact on the other. When an agent starts up again, it simply starts from where it finished last time.