The replacement process to resolve the issue of a failed capacitor is a tedious repair, and any qualified service provider that has a modern workstation to repair logic boards, graphic cards, and other electronic may be able (willing) to consider it. The part itself can be relative inexpensive, otherwise. So you pay for the experience and also the time it takes to get to the point of accesing the area where the part resides in hiding...
A general search online located an example of such a capacitor, though the one in your device may vary from the one shown in this page http://www.thebookyard.com/product.php?products_id=1884
Some iBook G4 and white G3s had an issue with solder failure on the logic board and third party cures varied from re-flowing the solder, to simply wedging the graphic or other component into place with a non-conductive space filler (plastic chunk) so if you happened to have one of the suspect models where a re-flow of some other logic board components may be needed for longer term use, you could find a company who knows these computers (wegenermedia is one of several authorized repair service companies who can reflow Mac circuit boards with modern workstations) and have them check over the computer, and replace the capacitor.
While the computer is open for such a simple part, the labor is already expended to access other areas of the interior, so that would be a time to consider a hard disk drive, or a legacy SSD from OWC, or a superdrive upgrade, and other tests, inspections, etc if you sought to keep it running as a curiosity. You can still get rechargable batteries from a few sources, owc and ifixit show some, amazon may show others.
The complete logic boad swap may be OK if a suitable part is available, but the actual repair of yours may be less if the company who does the repair also takes yours apart to perform it. And if you need any other parts inside upgraded at the time, that can be a labor bonus, if those other parts are reasonably priced and installed at that time, with some guarantee...
Good luck & happy computing!