All you need to do is a create a rule.
When setting up, select “receiving SMS”.
You can select to forward only SMS coming to Shared Numbers or a particular dedicated number of yours. If you want to forward inbound SMS regardless of the number they are being sent to, pick “Any Number”.
You can then enter as many keywords as you please if you only want to forward particular SMS depending on the keyword(s) they contain. If you would like all incoming SMS to be forwarded regardless of keywords, leave these boxes blank.
Go to the next step, and select “Forward to Email(s)”.
A box will display. Enter all the email addresses you’d like to forward your incoming messages to. Remember to separate them with a comma.
Click Save Rule and… Voila! All inbound messages that meet these conditions will be forwarded to your Email address!