next time I need [something], I'll...
next time [clause], [sentence]
"Next time..." is a conditional clause that describes the sentence's verb happens. the clause itself doesn't necessarily happen, and you are not implying that it is going to happen.
that is why you can't use "will".
you could also think of it as a cause and effect clause.
like in math? P->Q? but not Q <-P
if [I need something] then [I'll...].
you aren't saying you will need something, you are saying what you will do IF you need something.
About the Helper Verb "Will":
if you know Spanish or a similar language, it's kind of like the subjunctive.
Using "will" or "I'll" implies that something is going happen. You know that it's going to happen.
If you use "will" while you are not completely sure, it implies that you believe that something will happen.
In your example, you aren't saying you "will need " you're saying "if you need [something]"
Examples:
I'll... < = this happens in the future for sure.
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I'll probably... < = CORRECT, likely to happen, you believe it will happen.
I'll might... < = WRONG, may or may not happen.
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If the sky falls tomorrow, I'll hide under my bed. < = if the sky falls
The sky will fall tomorrow, I'll hide under my bed. < = when the sky falls
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Next Time we get pizza, I'll get cheese. < = might happen
Next week we'll get pizza, I'll get cheese. < = this is definitely going to happen.