These recommendations are based on my experience in May 2017:
Camino Vegano - all vegan restaurant near the cathedral. Tell them you are a pilgrim, and the menu noche costs 12.50 euros instead of 14. Some things on the menu were not available, but we didn’t mind. Friendly staff, we were the only people there, but apparently there are many more pilgrims at lunch, and locals on the weekends. However, I can't find the listing in HappyCow anymore. It's still on TripAdvisor, though, with pretty recent reviews. One reviewer said the name had changed, but they couldn't remember the new name.
https://www.tripadvisor.co.uk/Resta...5-Reviews-Camino_Vegano-Logrono_La_Rioja.html
Café El Sol - vegan restaurant inside a bar; the tapas sold and displayed at the bar were not vegan or vegetarian when I visited, but everything on the sit-down menu was. However, HappyCow is now listing it as fully vegan. Mostly pizza, hot dogs and burgers, and also some pinchos and raciones. The burgers are very good, pizza not so much.
Begin Vegan - 100% vegan shop, near Café El Sol. Sells hummus, veg patés, toiletries, and the other usual stuff. Also some pastries, including fardelejos - a local specialty in La Rioja introduced by the Moors. Usually made w/ lard, but the veganized version is probably more true to the Muslim original. Wonderful owner, very friendly.
Barrio Bar - vegan-friendly tapas bar near Café El Sol, recommended by the owner of Begin Vegan. Didn't have a chance to go there myself.
Bar Cid - tapas bar that specializes in setas (oyster mushrooms). Delicious!
Taberna del Tío Blas - another tapas bar, had a croqueta of spinach with almonds and pine nuts. It was delicious and tasted really cheesy. Their main drink specialty is a granizado (slushy) made with La Rioja wine. Was one of the few open in early evening (6:30-7:30pm) for foreigners, though the kitchen doesn’t open until 8pm. Several vegan-friendly tapas marked on the menu.