"I'll lend it to you if you'll bring it back tomorrow."
Does this sound good? Does this sentence establish the chronology as the first action being the lending of something, the second action being the bringing this something back?
It does establish the chronology correctly and it's grammatical.
However, you could say the same thing omitting the contracted will in the second clause:
I'll lend it to you if you bring it back tomorrow.
Both are perfectly acceptable:
I'll lend it to you if you bring it back tomorrow.
I'll lend it to you if you'll bring it back tomorrow.
As an American English speaker, I favor the latter, but both are equally fine.
I'll lend it to you if you will bring (you'll bring) it back tomorrow.
I'll lend it to you if you bring it back tomorrow.
Both are possible.
In the if-clause, we usually use a present tense to talk about a possible future action. (if + present + will)
'If you bring it back' means you may bring it back or may not.
If you eat too much, you will get fat. You will get fat if you eat too much. (NOT if you will eat)
We can use will in the if-clause when we make a request. But the meaning is not really future.
if you will bring it back= please bring it back
If you will wait a moment, I will fetch the book (If you will wait a moment= Please wait a moment)
If it is fine, we will have a picnic. (NOT If it will be fine)
I'll phone her if I hear any news. (NOT if I will hear any news)