How many atoms of carbon are there in charcoal briquets used to cook?

Those briquets are a mixture of carbonized wood, coal, and sawdust. Given the heating value of those briquets and the enthalphy of combustion of plain carbon, it is a pretty good assumption to say that, at least, about 50% of them is carbon content by mass so, per 1 kg, you have

Coal Content in Mass = 0.5 kg

Taking a per-kilogram calculation. Now, the atomic weight of carbon is 12 kg/kmol, thus

Number of C moles = 0.5 / 12 = 0.0146 kmols

And, thus, the number of atoms is

Number of C atoms = Number of C moles x NA

so, you are looking at something of the order of 250E23 atoms per kilogram of those briquets. Don't know how much those briquets weight in average.

Let us assume that the entire briquet is made of carbon. I happen to know that it is not but rather it has quite a bit of oxygen, hydrogen, nitrogen and even sulfur in it (they do have coal in them) along with some other elements. Even so, we can assume anything we want. Now I have a kitchen scale that I have massed four charcoal briquets on and they have a combined mass of 96 g, so we can assume that on average each briquet is 24 g.

Knowing that carbon-12 has a molar mass of 12 g/mol, we can calculate that each briquet has 2 moles of carbon in it, or

somewhere close to 1.2 x 10^24 atoms of carbon. This is assuming of course that the briquet is made entirely of carbon.