I don't get the chance to speak in English often.
I've been told in doesn't need to be in this sentence, but does its inclusion make the sentence grammatically incorrect?
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The sentence is correct with or without "in".
It's subtle, but there is a slight difference in meaning between the two.
With "the chance to speak English", "speak" is a transitive verb and "English" is the direct object. The intent is to use English itself, like a student who wants to practice. Communication may not be important at all.
With "the chance to speak in English", "speak" is intransitive, so it just means "communicate", and the communication happens to be in English. An English native speaker who lives in a country where nobody speaks English might say this if they happen to meet a fellow English speaker because they're talking about the experience of communicating using English.
It is grammatical with or without the in.
In most cases we would not use the in, but we could do, especially when we were talking not about the person's ability, but about their choice at a particular time. I don't think there's a context where we must use in.
By the way: the title of your question is ungrammatical. You need either Is using "in" in the given sentence grammatically incorrect? or Does using "in" in the given sentence make it grammatically incorrect?. With a lexical verb "make" in base form, the auxiliary must be "does".