The strap is his bandoleer ^[1] (more commonly spelled "bandolier" and often just called a "utility belt"). I don't recall him ever specifically utilizing the contents in any of the books, but they typically contain items such as ammo, grenades, supplies or small tools.

In Chewbacca's case, it is most likely that his contains ammo and tools.

Unofficially, Peter Mayhew (the man behind the mask) explains on his own website ^[2]:

A bandolier slung from Chewbacca's left shoulder contains enough extra firepower to take on a squad of stormtroopers.

[1] http://starwars.wikia.com/wiki/Bandoleer
[2] http://www.petermayhew.com/chewie.html

It should be noted that it isn't just a bandolier: it is also the strap for a messenger-bag-style satchel he carries around. This bag is easy to miss, because it tends to sit low on his right hip (sometimes behind him, out of sight), and it is almost exactly the same color as his fur.

Thus, even if the compartments on the bandolier were empty or absent, it would still be functional, because it supports the bag that carries his other gear.

The best shot I can find of the bag:

I found this picture in a reddit post titled "What do you think was in Chewbacca's satchel?" ^[1], and in the first reply to the post, Peter Mayhew ^[2] (the man who plays Chewbacca)¹ explains what the bag itself contains:

Aluminum ammo boxes.
Cheers,
Peter Mayhew

And the novelization of The Force Awakens says:

THANKS TO THE snow and the heavy forest cover, the patrol droid did not see them, and the deformation warp from a heat distorter Chewbacca carried in a pouch served to mask their thermal signatures.

We don't know if the pouch mentioned in this quote is the one attached to his bandolier, but it seems likely.

Although I couldn't find any canonical information about the contents of the bag or the bandolier compartments, I did find this ^[3] detailed description of the bandolier and bag on a fan site page devoted to making Chewbacca costumes:

1. There are 10 large blocks and 9 small blocks².
2. According to Dan Hyatt, the dimensions of the blocks are:
   • Large block, main piece: 3 inches X 3.75 inches X 1 inches
   • Large block, base piece: .125 inches X 3.252 inches X 4.002 inches
   • Small block dimensions: 3.75 inches X 1.38 inches X 1.125 inches
   • Space between each block (regardless of block size): 1.38 inches
3. According to measurements made by Jeff Allen using a laser pointer caliper to measure the Magic of Myth exhibit, these are the dimensions of various items on the bandolier:
   • The small coin-shaped item is 1.1811 inches (3 centimeters) in diameter.
   • Large block: 1.9685 inches (5 centimeters) by 3.1496 inches (8 centimeters).
   • Small block: 1.1811 inches (3 centimeters) by 3.3465 inches (8.5 centimeters).
4. The blocks are allegedly old aluminum project/science experiment cases from Radio Shack. These items are discontinued.
5. The rings used to attach the bandolier to the pouch are approx. 1 inch in diameter. (source unknown)
6. Chewie wears the bandolier on his left shoulder with the pouch resting on his right hip.
7. There is a hidden strap that attaches to the bandolier at the shoulder and goes around Peter Mayhew's neck, to keep the bandolier in place.
8. The Revenge of the Sith pouch appears to be made of cloth and not leather.
9. The Revenge of the Sith bandolier doesn't have the leather concho/checker piece/coin like the Classic Trilogy bandolier.
10. The Revenge of the Sith bandolier has some border lines in the both the main strap and the center strap. There are also some small leather loops on each side of the block that are probably for holding the center strap to the main strap.
11. The Episode 3 bandolier has blocks without any lip, and they look like they were assembled kind of like a 3D puzzle.
12. The bandolier used in A New Hope and Empire Strikes Back has simple tapered ends at each end of the strap. The Return of the Jedi bandolier has a curved, more ornate taper.

---

¹ The redditor using the name Peter Mayhew is indeed the real deal - he has posted pictures ^[4] that show the real Peter Mayhew, and they seem to be the kind of photos Mayhew would have the best access to. And more importantly, the account in question did an AMA ^[5], and Reddit is pretty careful about verifying the participants in AMAs.

²Most costumes make 9 large blocks and 8 small ones, simply because using the full number of blocks at their actual size would make the bandolier come down to most people's knees. Unless you are actually 7'4" you'll want to shave off one of each block or scale the whole thing down for your size, or do both.

[1] https://www.reddit.com/r/StarWars/comments/2pzr1b/what_do_you_think_was_in_chewbaccas_satchel/
[2] https://www.reddit.com/user/PeterMayhew
[3] http://gotwookiee.com/#bandolier
[4] https://i.sstatic.net/4lwFE.jpg
[5] https://www.reddit.com/r/IAmA/comments/1ljlju/i_ama_peter_mayhew_chewbacca_from_the_star_wars/

As detailed in Star Wars: The Visual Dictionary ^[1], his bandolier boxes (shown below) contain ammunition for his Bowcaster.

This is nicely backed up by the Force Awakens: Visual Dictionary ^[2] which implies that the smaller boxes contain a single quarrel.

[1] http://starwars.wikia.com/wiki/Star_Wars:_The_Visual_Dictionary
[2] http://starwars.wikia.com/wiki/Star_Wars:_The_Force_Awakens:_The_Visual_Dictionary

Well, it couldn't be amno, because they let him carry it even when they're captured by the imperial troopers in Empire Strikes Back.

So I guess it's snacks.