Is there any open data about the extent of partisan division among the police or armed forces in the United States?
If such data does not exist, why might that be? For example, is there some law which prohibits gathering such data (if any)?
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For one thing there's this Gallup poll which headlines as "more veterans are Republican" but actually shows roughly equal numbers of Republican and Democrat registrations among the veteran population as a whole.
There is more of a "partisan gap" when you add active duty personnel in particular, and this is largely due to how age plays. Veterans, as stated above, are roughly evenly split - with older veterans even trending more towards Democrat.
The 18-49 bracket, however, is profoundly Republican. There's selection bias here, if you're young, male, and Republican you're probably also pretty angry and as a former military recruiter (whose name slips my mind right now) said: "We want 'em young, pissed off, not particularly attached to their home, and trainable."
Younger vets are also coming home from conflicts where existing military doctrines have not been serving them well. The U.S. armed forces are still learning their way out of being a set-piece battle force and into the more complex, fluid, and (compared to other conflicts the U.S. has been in) dangerous realms of counterinsurgency operations and the like. A lot of them are coming home with a host of very muddy experiences that leave traumatic prejudice in place and whatnot. The current mode of Republican politics plays to fit those perceptions very well. (Older veterans, especially WWII vets, had a VERY different battlefield experience.)
As for laws that make it harder to measure this stuff:
This summary of the rules/regs that govern DoD personnel's political activities explains it very well:
The limitations of participation can be found in DOD Directive 1344.10, Political Activities by Members of the Armed Forces, and the Hatch Act.
Under DOD Directive 1344.10, members of the armed forces who are on active duty are permitted to express their personal opinions on political candidates, make a monetary contribution to a campaign, sign a petition to place a candidate's name on the ballot, and attend a political event as a spectator. Members on active duty may not participate in partisan activities such as soliciting or engaging in partisan fundraiser activities, serving as the sponsor of a partisan club, or speaking before a partisan gathering. In addition, all military members, including National Guard and Reserve forces, are prohibited from wearing military uniforms at political campaign events.
The Hatch Act allows most federal employees to actively participate in political activities on their own time and outside of the federal workplace. There are, however, significant restrictions on fundraising, running for office in partisan election and using one's official authority in the political arena. Senate-confirmed presidential appointees and non-career SES employees are subject to additional limitations. Additional restrictions are also applicable to career members of the Senior Executive Service.