Is this a vapor lock issue or something else? How expensive would it be to fix?

This symptom is usually caused by an issue with the evap system.It solely depends on vehicle type,but this system usually includes the gas cap,various hoses,a rollover/ vent valve,carbon canister and a purge valve.This system basically prevents fuel vapors from escaping to the atmosphere and these vapors are instead sent from the fuel tank to a carbon canister where they're stored until the engine uses a purge valve and engine vacuum to draw these vapors directly into the engine usually through a intake manifold vacuum fitting where these vapors are then burned just like raw fuel is.So it's part of the emissions system. When these vapors aren't allowed to escape from the fuel tank via the system components I listed,it basically creates a vapor lock condition and the symptoms you're having.It could be something as simple as a faulty gas cap or leaking/ clogged evap system vacuum hose or possibly a bad vent valve or purge valve.Most of these components are easy to access and replace with the exception of the vent valve which is located in the top of the fuel tank,which would obviously require you to either drop or,hopefully,just lower the fuel tank to access/replace the valve.If a check engine light comes/stays on when the symptoms are present,a code would possibly be stored which you could verify with a scanner.If you don't own one,AutoZone/Oreillys/Etc will scan the vehicle for you also if you don't own one & it's free.

It could be an emmissions problem. Sometimes even the gas cap being loose can cause a code. Make sure to tighten the cap when done refuelling...you should hear repetitive clicks when its tight. Other components related to code p0172 are o2 sensor(bank 1) and catalytic converters. Hope this helps!

There are normally two hose lines from the fuel filler that run to the tank. The larger one carries the fuel from the nozzle to the tank. The smaller one is a vent hose that allows vapors to leave the top of the tank while filling. These vapors come back to the nozzle area and can be vented and/or collected by the nozzle depending upon air pollution standards in your area. If your second hose is running to the bottom of the quarter panel, it is probably the vent hose that may not have been reconnected at the top of the tank when the fuel pump was replaced. The fuel running down there is from kickback fuel at the time the pump shuts off. The real problem is that the vent line is open on top of the tank and when the tank is full, that can allow vapors and fuel to escape out of there and it could contact hot exhaust areas - a major fire or explosion hazard. Have this inspected and repaired IMMEDIATELY.

If this is related to the fuel filling, evaporative canister issue, maybe the following will help. Most of the components for the fuel tank are located near the tank itself, with some near the front of the tank and the evaporative canister components behind the driver side rear wheel inside a splash cover.

The biggest problem people complain about is not being able to fill the tank without the pump stopping if the tank is filled at a higher rate than a trickle. This issue is caused by trying to overfill the tank after the fill nozzle has clicked off causing fuel to damage the evaporative charcoal canister clogging it and causing the blow back to the filler neck and fill nozzle, simulating a full tank shut-off of the filling process. This is not exclusive of Hyundai, and gas pumps usually warn of overfilling the fuel tank because of this known result. This evap containment mechanism was developed for the cars to replace the vapor collecting mechanism that gas stations used to use partly for the fill nozzle and collection dangers, now all that vapor is collected and safely burned when the engine is running. The evap canister can be expensive to replace, in the neighborhood of $150 from the dealer, and it's not recommended to try and bypass the system or you can create more issues than you solve.

Visit https://www.hyundaitechinfo.com/ to get service info for your vehicle, it's free and very useful.

If this isn't related to your issue, sorry, maybe you can post more info on what the issue is and how you need help.

If it runs good for 20 miles then stalls...its because something is heating up. I would check the catalytic converter.The cars old enough for it to be going bad....Its in front of the muffler. Also check to see if your cars vapor locked. Open your gas cap and if your hear it sucking in air this might be the problem as well. Just loosen the gas cap and see if it still stalls. You can drill a tiny hole in center of gas cap. It will be a quick fix. To fix it right replace the vapor filter canister! Its clogged from over filling with gas. That extra 25 cents run down the gas vapor line. Your not only filling up your tank..but the vapor line too...Round to the nearest nickel.

It may just be the fuel pump seal at the top of the tank. it has to be dropped for a definite diagnosis. It is rare but the tank seams occasionally crack too. Good luck.

You either have an obstruction in the filler tube or the vent tube on the filler is clogged so it is not allowing air to escape as the gas is going in.

Try filling up a little less. To me it sounds like a vapor lock issue which clears after you burn off the top bit of gas.. Next time you need fuel up be sure to keep track of how much you put in. Another idea is to run the tank down and then only fill up to half a tank. If the problem clears then you know that you're having a vapor issue probably due to gas saturation of the canister (which can happen if you overfill the tank).

Fuel tank is not venting proper, most likely vapor recovery system has a blockage.

I have a 2000 camry that I just fixed today. there is a vent tube and valve on the carbon canister that is located just behind the gas tank and above the cross member. this vent tube will get plugged up by ( of all things ) spiders, dirt daubers, and I even had a mouse pack the tube with dog food. Just remove the tube and the valve from the canister and clean them out. You will now be able to fill your gas tank again.