By law, you can ship a firearm of any type to an FFL in any state. Shipper policies are the current problem. Currently, only USPS accepts firearms from non-licensed persons for shipment, and they cannot ship handguns, only long guns.
You need an FFL that has a shipping account with UPS or FedEx to ship handguns. ShipMyGun.com provides that as a service, you need to take the package to a UPS facility, not a drop point or UPS Store or the like.