The simple answer is yes; the more accurate answer is, it depends.
Most of the missions are active catholic churches. They require that either the bride or groom be a practicing catholic. A few of the missions (Santa Barbara, for example, the last time I checked) limits weddings most of the year to members of their parish, because the demand is too great.
Unfortunately there is no standard procedure for arranging a wedding service. A few missions have instructions you can access on line but in most cases you have to start by calling the parish rectory (often a different number from the mission / mission gift shop).
Some missions are state parks and these are open to all faiths for services but usually require more advanced planning and some flexibility on when they are available, particularly on weekends.
A key consideration of course is which mission(s) offer the best setting? In the immediate (if there is such a thing) LA area San Gabriel has a splendid church (the congregation is largely Vietnamese) but limited grounds. San Fernando Rey has expansive grounds but the church exterior is rather plain and not very mission like, (the interior is beautiful). If it is feasible to hold the service further away from LA I would consider Santa Ines (in Solvang), a beautiful mission with splendid grounds or San Juan Capistrano, the most romantic of the missions (the only church on the mission grounds is the small Serra chapel and it might take some pull to make that available. I am told they like to use the modern church nearby). If anywhere is possible I would consider San Antonio de Padua, an exceptionally beautiful mission and the only one with its original setting still intact. There is a charming small hotel nearby. Both facilities are surrounded by government land which is why they are protected.
If neither you nor your intended are Catholic La Purisima would offer a great setting.