Carrying hydrogen gas directly would be difficult, and perhaps not useful. In real biology, the oxygen isn't carried as a gas, but as oxygen radicals attached to the iron atoms in the heme molecule.
However, it is possible. Your best approach would be to use a chelated compound with palladium at its core.
You can think of a chelated compound as a sort of organic "claw" molecule that "grabs" and holds a metal ion. The claw part is called a ligand.
These kinds of compounds are very common in biology, actually. The aforementioned heme (part of hemoglobin) uses iron as its metal, which of course is why blood is red (it is literally rust).
Similar compounds include hemocyanin, which uses copper as its metal instead of iron. This gives blood made of hemocyanin a blue color, as opposed to red (hence the name).
Another example is vitamin B-12, which uses cobalt as the chelated metal.
In chelated compounds, the metal is generally the active site for the molecule as a whole, and the ligands are used to control the reactions associated with it. You can think of the metal as a kind of "power core" in this case.
So we've established, based on real biology, that you most likely want some sort of chelated compound. So why palladium? Palladium is a precious metal of the noble metals group. It has the interesting property of acting like a "sponge" for hydrogen atoms, being able to pack them much closer together than in their gaseous form.
However, the use of palladium and hydrogen here over iron and oxygen introduces its own problems:
In the two heme examples, the molecule is designed to carry highly reactive oxygen radicals that can be passed on to other proteins for chemical reactions. They aren't carrying oxygen gas directly.
Secondly, palladium's "sponge effect" comes from the lattice structure of its metallic crystals. This means you need not just a single metal atom or ion, as in the previously mentioned cases, but hundreds, probably thousands, of palladium atoms in crystalline form. This won't fit into a simple ligated molecule.
Another problem with palladium is that it requires heating to release the hydrogen it's captured. Your creature will have to have differing temperatures across its body; essentially a "cold section" where it breathes in the hydrogen, and a "hot section" where it releases it.
The last problem is palladium's rarity. As a dense and heavy metal, it would sink to the core of any planet its on, and not easily be available to life to use. On Earth, it's about 3 times as common as gold, but iron is literally millions of times more prevalent. Palladium is present in Earth's crust at 15 parts per billion, but iron is present at 63 parts per THOUSAND. There is on average literally more iron in the crust of the Earth than there is salt in the ocean (about twice as much). This will make it difficult for your creatures to find and consume in any kind of meaningful biocycle.
But your question is "how would it work," not "would it work," so let's try and create a scenario where we can use it:
Your creature's blood would need specialized cells, these cells contain small granules of palladium, let's call them "pallophores." Rather than lungs, your creature could breathe through its skin, which is covered thousands of tiny ruffles packed with pallophores in order to maximize surface area. This effectively increases the surface area for respiration, allowing your creature to pump its blood slower. The fact that the pallophores are pumped close to the skin surface allows them to be better cooled for hydrogen uptake.
Because your creature's pallophores are quite literally like little grains of sand in its blood cells, they couldn't be very flexible, and so your creature would need to have relatively large blood vessels, especially capillaries. The tradeoff is that you could get more density of hydrogen atoms packed into the pallophores than you would be able to for oxygen packed into heme.
Now, we need a way to get the hydrogen back out of the pallophores. Probably the easiest option here is for the creature to have "nodes" in its blood stream where pallophore rich blood cells gather, perhaps little sacs of some sort. These sacs are heated, allowing the hydrogen gas in the pallophores to diffuse out into the surrounding tissue. Your creature will need a lot of these pallosacs in order to provide for its hydrogen needs.
The hydrogen gas is reduced with some other chemical, presumably NOT oxygen, into an easily removed (preferably liquid or ionic) waste product. This waste is carried through a second blood loop where it's excreted by a kidney-like organ.
Because of all the biological limitations we've had to place on the creature, its metabolism will necessarily be slow, so in order to survive, the creature must be large and slow. We can use this ecological niche to solve our rarity problem: Let's assume that there are some kind of plant-like organisms that pull palladium out of the soil and concentrate it in their bodies, and your creature eats the plants in order to get the palladium.