We develop projection operators onto finite element differential forms over simplicial meshes. Our projection is locally bounded in Lebesgue and Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover, it incorporates homogeneous boundary conditions and satisfies a local broken Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond projection and a modified Cl\'ement-type interpolant with the projection property. The latter seems to be a new result even for Lagrange elements. This projection operator immediately enables an equivalence result on local- and global-best approximations. We combine techniques for the Scott-Zhang and Ern-Guermond projections and adopt the framework of finite element exterior calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini, N\'ed\'elec, and Raviart-Thomas elements.