Do any quantum interpretations not include nonlocality? Are there any interpretations of quantum theory that have a mechanism such that there is no need to invoke nonlocality?
 A: There's no necessity for the nonlocality in the quantum theory. In fact all experiments to this day can be explained by the local interactions.
The myth about the "quantum nonlocality" comes from the impossibility to explain the experiments with help of the local hidden variables. I.e. if you think that the quantum theory is not complete but the approximate description of some classical reality then you have to introduce the superluminal interactions between these classical degrees of freedom or to make them nonlocal. If you don't try to fit the quantum physics into the classical paradigm, no such problem occurs. The "Copenhagen interpretation", various talks of the "many worlds"/"many minds" when they are done right, consistent histories based interpretations don't involve any nonlocality at all.
However the hidden variable people made quite a lot of popularizing their language that presents the problems with their approach as if they were the problems with the quantum physics itself. This presentation is sadly widespread in the popularizations of science. It is also popular among the researchers on "quantum foundations" because it allows them to present the experiments (sometimes useful because of new tricks in their realization) that predictably confirm the quantum physics as if they were at the forefront of the modern physics.
A: I think so. In my work http://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-013-2371-4.pdf (Eur. Phys. Journ. C,  (2013) 73:2371) I consider scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics) and spinor electrodynamics (Dirac-Maxwell electrodynamics), show that the matter field can be algebraically eliminated in a certain gauge, the resulting equations describe independent evolution of electromagnetic field and can be embedded in quantum field theories (see references there to other people's results that I used).
To understand how we can do without nonlocality, we need to understand why nonlocality seems inevitable. People say that it is a consequence of the Bell theorem. The latter contains two statements: 1) local hidden-variable theories satisfy some inequalities; 2) these inequalities can be violated in quantum theory. However, the proof of the second statement uses both unitary evolution of quantum theory and the projection postulate, which are, strictly speaking, mutually contradictory (for example, it is shown in https://arxiv.org/abs/1107.2138 (Phys. Rep. 525 (2013) 1-166) that the projection postulate is only an approximation). On the other hand, experimental demonstrations of violations of the Bell inequalities have some loopholes. The "loophole-free" articles of 2015 assume that measurements have definite outcomes at a certain time point, but this assumption is, strictly speaking, not compatible with unitary evolution. . 
