The way all these things work is you can set up a webhook. (A webhook just means, “when X happens, we’ll execute an HTTP request somewhere.” Typically, you’d set that up as a POST to a backend workflow endpoint in your Bubble app.)
So, in SendGrid, this feature is called Inbound Parse. (SG parses the email, and sends you all the info via webhook.) An introduction is here:
Any other system works very similarly and typically the interfaces are more or less identical for SMS and email.
While I didn’t go into detail on this (this is so easy that it hardly bears explication), I described my experience setting up inbound SMS parsing in ClickSend here:
Hope that helps.