Yes - you can use a reaction on your turn to cast this if you have one available
A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else's.
Since hellish rebuke takes a reaction to cast, it obviously qualifies.
If a spell can be cast as a reaction, the spell description tells you exactly when you can do so.
The spell specifies that it can be cast
in response to being damaged by a creature within 60 feet of you that you can see
A common cause of getting damaged on your own turn is getting hit by an opportunity attack as a result of your movement1. This would be a valid trigger for the casting of hellish rebuke.
So, if it is your turn and you get damaged and you have a reaction you can cast hellish rebuke as a reaction.
Note: No bonus action spells on your turn before or after hellish rebuke
Note that this means that you cannot cast a bonus action spell on your turn after casting hellish rebuke. If you have cast a bonus action spell already, you cannot cast hellish rebuke during your turn at all2. However, casting a bonus action spell on your turn would not affect your ablilty to cast hellish rebuke as a reaction outside of your turn. This is confirmed by Jeremy Crawford here.
Note: This means you spend your reaction
If you use your reaction on your turn, you are not able to use another one until the start of your next turn3.