Any way to find a seller by partial name?

No, that's not possible Yuriy. You can view a list of recently updated profiles, but this changes constantly so not really suitable for what you want. There is also a search box on that profiles page, but it's not a search, just a lookup, as you need to input the members exact membername for it to work.

Trade Me Member Profiles | Trade Me

If you have dealt with them previously you can search your emails using the partial info you have, or if they regularly sold certain items you could try searching for the items on Trademe and see if you can recognise them.  I often search, see a photo and recognise who the seller is without going in to the listing.

The only way to browse alphabetically is in your saved Favourite members list, so if there is someone of interest for the future, you could add them there and just set the email new listings frequency to none. Best of luck.

Yeah, simply save your users as favorites is the best. Thank you!