wheelchair rental in zurich

These are my best search results:

http://www.alohatransport.ch/en/wheel-chair-rental/

http://www.mis-ch.ch/en/home/ - I didn't find anything good, but you could contact them.

I'm guessing that residents would get wheelchairs through the national health service.

You may want to contact your hotel and ask - normally some larger pharmacies would have them or be able to direct you to the right place. This pharmacy for example is fairly central and rents them (Rollstuhl = wheelchair, Vermietung or Ausleihen = rental)
https://www.apotheke.ch/limmatplatz/dienstleistungen/hilfsmittel-mietekauf

Might be difficult to read this if you don't speak German, but you also could try Spitex (home rehabilitation and medical service) as I know they rent wheelchairs and other equipment and have locations in Zurich (Aussersihl Hard is closest to you)
http://www.spitex-zuerich.ch/standorte/krankenmobilien-magazine/

Hope that helps (and by the way there is no nationalized health service in Switzerland!)

Perhaps call the local tourism office as they will speak English and may have encountered this before.
Like Hopper said, no national health service here. I think that's the UK and Canada. Ours is similar to the US Affordable Care Act (Obamacare).