There are direct Eurostars between Avignon and London in July and August only. At other times of the year, your best option is to change at Lille rather than going into Paris and having to cross the city. Look for fares on www.eurostar.com but also compare prices for separate legs from Lille to London at www.eurostar.com and Avignon to Lille at www.loco2.com and www.capitainetrain.com (these are relatively new resellers that are much better for US residents than the official site is).
Remember that while you can book tickets for Eurostar six months in advence, French domestic trains don't go on sale until three months in advance. Also, if you book separate tickets, remember the thirty-minute check-in time for Eurostar, I would suggest leaving at least an hour to change trains at Lille in case.