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Let $U\subset X$ be an open immersion of separated algebraic stacks of finite type over a field , Let $E$ be a vector bundle on $U$ .

In which cases can we extend $E$ to a vector bundle on $X$? Same question if $U$ is a Deligne Mumford stack or a quotient Deligne Mumford stack.

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Forget stacks -- even for schemes this is a serious problem (especially since you mention no smoothness hypotheses). Please tell us what sort of answer you find to be satisfactory in the scheme case, so we have a sense of what you're looking for with stacks. –  user36938 Jul 25 '13 at 3:11
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