It all depends on how much concrete is required. If it's just a wheelbarrow full you could almost do it yourself. In any case you'd have to get some estimates, but if it's just mixing and pouring a little concrete, I can't imagine it would cost more than a couple hundred bucks if you find the right contractor.

I had issues like this around my driveway and front porch. Do you have clay soil? Ours took 30 yrs to fully settle around the house. It sunk on 3 sides of the house and I had to fill in so water would drain away. Under the porch, I stuffed dry soil in there as much as I could. It would settle again. When the gap got too small to reach in, I used a piece of plastic hose stuck back in there to funnel thick mud slurry. Several times over the years. I had possums and other critters living under there. Finally got it filled in well enough they don't get in anymore.

However if there's anything really important like a driveway that has to take the weight of cars, or the house, a gap next to the foundation can cause a big crack, settling, cracking walls, etc. Turned out I had a strip all the way along the front of the garage extending 2 ft out from the garage slab that had a gap under it almost a foot deep. The driveway had sunk 2" and was probably in danger of breaking off. I had a foam contractor fill it in, which worked great and raised the driveway. However it took a lot more foam than initially estimated which raised the price from under $2000 to $3000. I wish I had just used a concrete saw to cut the strip of driveway off, filled in with compacted gravel, and hired a good concrete guy to pour a new strip to cover it. It probably would have cost 1/4 what I paid. Live and learn.

Houses may shift on clay soil or settle more than on other soils. I was advised by a structural engineer to always water the foundation on a regular basis. That does not mean to spray foundation with water, place sprinklers or drip hoses about 2 feet from foundation and water good off and on. House settling or shifting can cause sheetrock to crack on the ceilings and walls.