I don't think anarchists have any obligation to support boycotts, and yet I do quite frequently, but with no illusion to that being an explicitly anarchist action.
An example: When I purchase coffee, I choose to purchase it from places that are not Starbucks for a variety of reasons (greenwashing their relations with coffee growers, exploitation of their baristas through just-in-time scheduling, their role as a force in gentrification/homogenization of the world, generally bad coffee, I could go on). Due to my job, I sometimes find myself in places where the choices for acquiring my caffeine fix (and we should be honest about that part too...) are limited to gas stations, the golden arches, or the green mermaid. In those instances I tend to choose the latter, and I feel not even a little bad about it.
I know anarchists who feel strongly about maintaining boycotts against the Very Bad Things. I also know anarchists who think that boycotts are pointless and don't take them into consideration ever when they are buying things. Here is the rub: boycotts can be effective, even if not-explicitly-anarchist, especially in conjunction with other more antagonistic tactics of opposition. No one person is going to stop a business, industry or state by withdrawing support, but if large groups choose to withdraw support it can be impactful.
Certainly this is acting within the economic realm, and thus not-technically-anarchist, but it is also the world we live in. We are not pure snowflakes of anarchy, we are polluted by and immersed in capitalism. When we pretend that either we are above any considerations of how our choices work in the real world because the shitty tray of options available to us doesn't match our ideal menu we are being disingenuous. Conversely, when people act as if not participating in a boycott is tantamount to collaboration, that ignores the reality that all of us compromise our ideals all the time.