I'd buy a prebuilt frame at least. The things have to be lined up just so or they'll eat up tires and steer badly.
Best case: Brand new Ural Retro (these have slightly smaller rims than stock, and I /think/ people have gone darkside on these. But they are /really/ expensive and don't have 2wd). Very attractive. Bike comes with leading link front end, sidecar all set up, has reverse. Parts are crazy expensive. Air cooled. :/ Splash oiler. :/ Pushrods... :/
Next best case: Brand new Cozy sidecar on EN500. Lightweight. New. Fiberglass. Attractive.
Most realistic cheap case maybe: Thrashed frame and tub from a dnepr or a CJ750 hooked to the EN500 (though if you end up with one of the 2wd setups, you'll have to hack it to be 1wd, though I expect this would only involve cutting the axle off. Cheap(ish). Heavy. Solid steel. Attractive if you like ugly utilitarian stuff. May be too heavy for the bike (you need a heavy bike for a heavy hack, light bike for a light hack, or the steering is crazy).
Least realistic cheap case: You fabricate one from scratch. If you've got the skill/patience to do this, you'd probably be better off building a trike with a light pickup truck or jeep axle and building a special doggie seat into it.
If I were you, I'd read more about sidecars before deciding that you really want one. There's a reason you don't see them anymore. They're slow, because if you go fast, you'll either fly the car over and wreck of nose over and wreck. The center of gravity is way off, and you have to throw your body weight all around to keep them under control. They have cool factor, sure, but unless you'll be keeping it under 60 mph on fairly untrafficed roads and have forearms like Popeye, I'd seriously consider a trike or towpac/voyager kit with a doggie seat and/or big trunk/trailer first.....