So I created a font using calligraphr and I added left and right quotation marks but the right does not type. I can add them using glyphs in Adobe Illustrator but it does not work in Photoshop. I see it in the font book app (Mac OS) so it is there. I just don't know how to get it to automatically type, and for it to work in Photoshop.

No idea about the PS version you're using.
Maybe you'd like to watch this video? https://www.youtube.com/watch?v=mWQoTqp9paU
or read this https://creativepro.com/photoshop-quick-tip-applying-opentype-alternates/