Allowing a client to choose a font

I really like @Test Screen Name's approach. Prepare a dummy document that shows fonts that go together. (My designer always did like that... and they regularly chose fonts that were not included in Adobe's font portfolio, so that we needed to buy them, instead of simply downloading... 😄 ). As you are the designer, do yours a favour and chose one of Adobe's (or Google's...)

A different approach. Most clients may not welcome being told to explore a web site with thousands of fonts, organised in ways they may not understand. Also, given a free rein, they may make choices that we, as a designer, know are not ideal for the look of the piece. They might really like Comic Sans and you might not. They may be wanting a funeral brochure and not realise that the fonts they use send a message of playfulness. They may choose an old fashioned font to sell a new gadget. They may choose a titling font for body type or vice versa.

So instead, you could could make your own list of candidates, with some notes about how they would affect the vibe and perception of the client's work, and deliver it to them as a PDF made in InDesign.

If they create a free CC account, can they browse TypeKit fonts? I have not tried this so I don't know if it will work.
If that isn't an option, you might consider a different source for fonts such as Google Fonts which is free and open source and has high-quality fonts for download.

https://fonts.google.com/