A repertoire consisting entirely of gambits; is it possible?

Sure it is...as long as youre serious about not improving.  

why would you not improve?

You can't play only gambits but you could play a lot of them with 1.e4. There's stuff like the king's gambit, Alekhine-Chatard, wing gambit and others. Otherwise there are plenty of ways to get at least exciting positions on the board. And contrary to certain opinions, there are plenty of sound gambits.

I think playing gambits frequently will improve your chess if you have a genuine wish to improve. If you're going to play them you must play vigorously, keeping the initiative and looking for moves and plans that maintain your attack. If not your material disadvantage will quickly count against you. Therefore you are forced to search for creative attacking ideas. Sometimes it won't work, but the process of constantly searching the board for dangerous moves will help you improve IMO.

Might be a fun way to learn to play attacking chess, that's for sure!

Danish against e5

Smith-Morra against Sicilian

BDG against Scandinavian and Caro-Kann

Miller against French

Pirc/Modern ???  there are gambits, but not really in the opening.  Grand Prix style

Eugland against d4

Scandinavian against e4

From against Bird's

d5 and/or b5 against c4

Many (sound) gambits against everything.  

Yeah, it's possible. Benko Gambit vs. d4, Scotch, Morra, Cochrane gambits vs. the most common e5 responses, etc. 

So it's possible for the majority of your openings to involve a gambit. 

If your opponent avoids a gambit by playing something maybe hypermodern, you can't really force a sound gambit, which is entirely different from dropping material every opening. 

I thought about this against the Caro-Kann but I don't really like it after 1.e4 c6 2.d4 d5 3.Nc3 de 4.f3 e5 (4...Qb6 5.fe e5). Is there another way to do this?

My repertoire is composed of a lot of gambits: King's Gambit,Urusov Gambit, Queen's Gambit(still studying),Benko Gambit,Icelandic Gambit,Adelaide Countergambit,Colorado Gambit and some more

queens gambit is not really a gambit ;/

Benko against d4 is the soundest i dont know a good one against 1.Nf3/c4 tough

The Anglo-Scandi ;)

is bad?

 If you plan to only do gambits, I recommend Alterman's Gambit Guide series. Most of the gambits are completely sound.

Check out....

Attack with Black by Valery Aveskulov.  The operative word here is ATTACK!  This features a gambit repertoire for Black against all of White's tries beginning with 1.d4.  The repertoire is based around the Blumenfeld, Vaganian and, primarily, the Benko, gambits. An excellent companion to this would be The Alterman Gambit Guide - Black Gambits 1 by Boris Alterman, which instructively treats the same gambits.

The Alterman Gambit Guide comprises three volumes:

The first volume covers White gambits (G.) as follows: Danish G., Urusov G., Cochran G., Evans G., Morra G., and the Milner-Barry G.. Several other openings are covered including the Morphy Attack vs the Two Knights Defense, the Max Lange Attack and the Panov-Botvinnik Attack. The second volume features Black Gambits against 1.d4 and 1.c4, primarily the Benko G., Blumenfeld G. and Vaganian G.. The third volume continues with Black Gambits against 1.e4 and 1.f4 - The Marshall Attack, Hector G., Traxler Counterattack, Frankenstein-Dracula G., Falkbeer Counter G., and From's G. See also Open Gambits by George Botterill.

A site devoted to gambits -

http://www.ianchessgambits.com/

I have been moving in this direction with the openings I play I used to play really solid stuff like ruy Lopez and Italian game, or queens gambit which is very sound and solid not even a proper gambit really. But I love exciting games they are so much more entertaining and interesting. I have spent months learning the kings gambit in response to e4 e5 and Smith morra gambit in response to e4 c5, in response to french I try out the reti gambit which is e4 e6 b3 d5 Bb2 and if they take on e4 accepting the gambit you get a lot of play and I have spoken to some of the strongest players at my club around the 1800 to 2000 fide mark that play the French exclusively against e4 and they said they wouldn't accept it white gets too much play against you and French players seem to be less used to the kinds of games you get from it. all of these stem from e4 on move 1. I haven't really looked at too many gambit from blacks side but I may start doing that soon. at the moment I want to focus on getting better at the kings and Smith morra gambit there is a lot to learn in them but when you're good with them they are incredibly strong openings, very sharp and they allow you to really put pressure on your opponents in a way that many non gambit openings don't offer and for anyone doubting gambits you have to remember that we are playing against human players not computers who cares if a gambit is sound or not from an engines analysis how many humans can play accurately in sharp positions where their opponents king is safe and they are the one who is suffering the full force of a dangerous attack which has the potential to lead to mate or huge material loss if they make even a single miscalculation. Pressure and psychology is huge in these lines.

You should consider to have non gambit sidelines as a backup if things go wrong (Like Ruy Lopez)

Not only will playing gambits hone your tactical and attacking skills, but it will also develop your appreciation for the importance of seizing and maintaining the initiative - i.e., TIME!  

Making every game into a gambit opening is not possible (well, unless you do so completely dubiously) but you can have a gambit/aggressive opening repertoire. On the club level the book Gambiteer 1 as white, and the books Attack With Black and Smerdon's Scandinavian as black seems like a good way to go about it.