Hi Elizabeth,
Sorry to hear that your JRT is having trouble with storms, it's a common problem. Yes, there are several very effective drugs that your veterinarian can prescribe to help your boy ride out the weather. However, the aren't any really effective over-the-counter drugs that will do the trick, and your veterinarian will need to do an exam on your boy before choosing the appropriate prescription for him. Some of the most common tranquilizer drugs can also lower blood pressure and so it's especially important for your vet to check out your dog for any problems associated with the cardiovascular system.
A few tips about giving tranquilizers if you know a storm is inbound. They tend to give the best results if given on an empty stomach, so you might want to skip giving him dinner if you think you might need to sedate him for an incoming storm. Most oral meds take a few hours to reach an effective level in the bloodstream, so if you're not sure if there's a storm coming, you're better off to give him a dose of meds and possibly have a "false alarm", than to wait until he's shivering and terrified and then try to sedate him. At that point, it's a lot more difficult to dose him sufficiently without resorting to injectable medications.
One more thing you can do to help lessen the effects of the storm is to stuff his ears with cotton, just as you would your own. Do NOT use any instruments to push material deep down into the ear, just what you can push in with your little finger is enough. Foam ear plugs are also helpful, the same ones you can purchase if your spouse likes to snore. Twiddle them between your fingers to squash them down to a smaller size, push them down the ear canal without using anything other than your fingers and let it expand enough to fill the canal and help block some of the sound. Dog's ears can be a little grimier than most human's (hopefully) so you might want to buy several sets rather than try to re-use the same pair.
You can also try turning on distracting sounds, like the TV or stereo, but don't try turning it all the way up in an attempt to drown out the thunder entirely---you won't be able to most of the time. your boy underneath bed covers with you seems to help calm down some pets that don't mind (or even enjoy) the closeness of blankets and might also help dim down the racket.
My last suggestion would be to take your boy to the vet for meds prior to storm season. Several of the available sedatives have a fairly wide range of effective doses and it's a good idea to be able to experiment a bit under non-stressful conditions so you have a better idea of what the best dose will be when the Perfect Storm comes into town.
Good luck!
Susan Garlinghouse, DVM
Dog Veterinarian
Animal Hospital Owner, many years owning and training