It varies depending on whether it is a known or an anonymous donor. In the latter case, it depends on the phenotype characteristics of the recipient: the more common they are, the easier for the clinic to proceed with the matching process.
If you choose a known donor, the process may take longer. Most sperm banks require recipients to pre-register with them, and then access to their database or catalogue is granted. Then, the only thing for you to do is choosing the most suitable donor for you. If you choose a friend or family member, it may take longer, as he has to be pre-screened.