Do I need SEMrush or Ahrefs to run an agency?

Not to report on a client's own website. Google Search Console is free and shows the searches, pages, clicks and positions for that site, and your server logs show which pages were asked for.

SEMrush and Ahrefs earn their price when a client needs data about other websites, such as competitor keywords or who links to a rival. If you need that for one client, rent one of them for that month. The cost article covers the simple stack in full.