L(M)=L where M is a TM that can move right or stay, so L is decidable
This inherently depends on your exact model of a TM, I'm assuming the following:
A left-bounded tape, the head starts at position 0 and the word to decide is written at the tape. Furthermore, the TM has finitely many states.

Now the TM can only look at the first symbol at a time, can do finite computation (i.e. can have finitely many transitions to different states (*)) and move its head to the right once. Then it can repeat itself (**). In that, this TM is quit limited in what it can do, it can do as much as an DFA (like pointed out in the comments), that is: Look at a symbol, do some state-transitions and look at the next symbol.

(*) What happens, if the TM loops in the inner states without moving its head? Well, since there are only finitely many states, the Universal Turing Machine UTM can simulate the TM and just count how many transitions the TM has made since the last head movement to the right. If this number exceeds the number of states of the TM, the TM will not terminate, so the UTM can answer 'no'.

(**) What happens if the TM just goes to the right indefinitely? Well, at some point it will exceed the word to decide and move its head to a blank cell. An easy argument is to just count from there on the number of transitions (moving right or staying with the head at the current cell). If this number is greater than the number of states of the TM, it is caught in a loop and the UTM can answer 'no'.

To sum it up: The TM in question cannot decide its language, but the UTM can, so it is decidable in the usual sense.