Here's my list of vegetarian-oriented restaurants. Not sure about which ones are closest to Sagrada, but hopefully google map will help.
Sesamo restaurant (requires reservations for dinner)
El Buffet Verde (vegetarian buffet)
Fresc Co (salad bar)
Catbar (bar with vegan options)
We were just in Barcelona and had a great vegetarian menu del dia at Arc Iris (I think it's also called Arco Iris, but the sign on the building says "Arc"). It's at the corner of Avinguda and Carrer de Roger de Flor, just a short, pleasant walk from La Sagradia Familia. They start serving at 1:00, so you have plenty of time to tour the cathedral in the morning and stroll to the restaurant. We're fairly picky vegetarians ourselves and were very pleased with the three-course-plus-postres offerings at a very reasonable price. We can't vouch for their dinners, but plenty of locals crowded in with us for lunch, so it was popular and not at all touristy.
I posted on here after returning from Barcelona end of June with reviews of a few vegie restaurants which I hope will help. sharonbetty
Here's the link if it works! Sin Carne is not totally up to date so don't rely on it.
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