Pulling up the carpet IS a pain in the ass and I would avoid it unless you know someone, (or want to pay a carpet install guy), to lay it back down all nice and neat, tucked properly at the edges/under the baseboard/on the tackstrip. You can really make things look crappy if your "carpet-laying" skills are as bad as mine were. I did it once, made a friggin' mess, had to call the "PROFESSIONAL" to come fix what I had screwed up.
However, depending on the length of your cable run, you might have success with one of those "electricians" fish tapes as Theo suggested. The big issue doing it that way is getting the leading tip of it slid under the carpet without jamming-up and snagging on the padding. You gotta make the cable go inbetween the underside of the carpet and the top surface of the padding. Don't try and get the cable under the padding, it's a disaster and besides, the padding is usually glued down to the subfloor. It'll be a frustrating mess. I ended up doing 2 things to insure success: (1) just spend a few extra dollars and call the carpet guy to lift up, and correctly re-install the carpet. You'll be glad you did. And (2) use good quality cable right off the bat. Don't be skimping, using cheaper shit just to save some cash. If you're anything like me, you'll end up wanting to tear the "bargain" cable out and replace it with another one, (which I did 4 times in a 10 year period), and that's a pain. Depending on whether you plan to run the sub line leval? or speaker leval? you'll want to go with FLAT stuff. I have used Analysis Plus, and Nordost. Both worked well. Both have many different "levals" in their lines, so you wouldn't have to sell your firstborn, or auction off one of your body parts to own a high quality subwoofer cable! And the fact that they are FLAT cables means your carpet will look less stupid, or bumpy and bulged-up where the cable runs. (Things like that matter in my house 'cause my lovely wife can't bear the sight of wires the size of garden hoses running across the floor). As for going wireless? I personally do not like wireless subs, Had one once, seemed to have issues with occasional signal loss and interference. Maybe something was wrong with the transmitter module? But I didn't hang on to it long enough to figure it out. Dumped it less than a month after buying it and went the Wired route, never looked back. Good luck with your installation.
However, depending on the length of your cable run, you might have success with one of those "electricians" fish tapes as Theo suggested. The big issue doing it that way is getting the leading tip of it slid under the carpet without jamming-up and snagging on the padding. You gotta make the cable go inbetween the underside of the carpet and the top surface of the padding. Don't try and get the cable under the padding, it's a disaster and besides, the padding is usually glued down to the subfloor. It'll be a frustrating mess. I ended up doing 2 things to insure success: (1) just spend a few extra dollars and call the carpet guy to lift up, and correctly re-install the carpet. You'll be glad you did. And (2) use good quality cable right off the bat. Don't be skimping, using cheaper shit just to save some cash. If you're anything like me, you'll end up wanting to tear the "bargain" cable out and replace it with another one, (which I did 4 times in a 10 year period), and that's a pain. Depending on whether you plan to run the sub line leval? or speaker leval? you'll want to go with FLAT stuff. I have used Analysis Plus, and Nordost. Both worked well. Both have many different "levals" in their lines, so you wouldn't have to sell your firstborn, or auction off one of your body parts to own a high quality subwoofer cable! And the fact that they are FLAT cables means your carpet will look less stupid, or bumpy and bulged-up where the cable runs. (Things like that matter in my house 'cause my lovely wife can't bear the sight of wires the size of garden hoses running across the floor). As for going wireless? I personally do not like wireless subs, Had one once, seemed to have issues with occasional signal loss and interference. Maybe something was wrong with the transmitter module? But I didn't hang on to it long enough to figure it out. Dumped it less than a month after buying it and went the Wired route, never looked back. Good luck with your installation.