Some European manufacturers of propane tanks recognize that it is inevitable that you may wish to try and fill them abroad. They may provide adapters for use in different countries. Ensure you are offered the newer style US adapter and not the old style. You will still have to persuade the US filling station to use the adapter - depending on the operator's approach, this may be problematic. The US does not operate a national tank trade-in system and those that do exist tend to be locally based.
Rental tanks are also rare.
Virtually every hardware store carries tanks and regulators. The regulator is usually available with a pigtail of low-pressure pipe that could be connected to your vessel's existing propane system. You should ensure that you take the new bottle to a reputable filling station - the first fill requires that the tank is air purged and leak tested. Repair the leak or if there was no leak, shut off the tank and gas consuming appliance. Wait for any gas to dissipate from the area.