While there is no general rule limiting the number of times the verb "will" can appear in a sentence, there is a problem with your example. Look at it this way, as written your sentence is two command actions (I will if you will!) It reads oddly to most English speakers because, well, without context "them's are fightin' words". It's considered impolite to compound commands.
The most common way to express the original sentence, as already pointed out, is:
"I would appreciate it if you would send me my bag."
Rather than a double command, this is a conditional-consequence pair (if this then that, or... if send then I would appreciate). This is considered polite as it indicates an obligation would be incurred (to show appreciation) if the action is taken.
(Note that it is improper to say, "I would appreciate it if you could send me my bag." Unless there is a reason to question the sender's physical ability to send the bag (could) the request is always one of permission (would). See "can I..." vs. "may I...". If you really want to be sarcastically funny, reverse the would/could pair... Suddenly the sentence is quite rude!)
Another way to say the sentence is:
"I will appreciate it if you do send me my bag"
This is another form of the conditional-consequence pair, but the use of the initial command (will appreciate) implies sarcasm (playful or angry). It suggests the sender can't be trusted to follow through with his or her action whether he or she agrees or not.
To bring us full circle, let's give context to the moment, such as the situation of two people daring one another to take a possibly risky action such as jumping into a pool for the first time:
"I'll do it if you'll do it!"
So, you see, the question isn't "can I use 'will' multiple times?" but "should I use 'will' multiple times?" And that depends on what you're trying to say and why you're trying to say it.