Your attempt is grammatically correct.
To specify the target of 「させる」, we use the 「に」 particle.
But your sentence can be improved a little more, so let me describe it.
First, 「質問を聞く」 is a little bit unnatural. We say 「質問をする」 far more often.
We use 「聞く」 with concrete question phrases, for example: 「どこに居るのか聞く」 (ask where you are).
Then your sentence would be 「先生が学生に質問をたくさんさせてくれた。」
Secondly, this sentence is ambiguous because there are two patterns in which 「学生に」 modifies different verbs; one is "let" and the other is "ask."
The two possible interpretations of the sentence are:
The teacher let the students ask lots of questions.
The teacher let (someone) ask the students lots of questions.
One way to avoid this is:
The teacher gave the students a lot of opportunities to ask questions.
In this case 「学生に」 modifies 「くれた」 but not 「質問」, because we don't say 「学生に質問」 to mean "the questions to the students" but say 「学生への質問」.