I'm not sure of the situation in the United States but in the UK (where I live) it could make a difference depending upon the type of career. In some professions - say researcher, librarian, accountancy - your medical history should not even be available to any interview panel. In other professions, psychological profiling can form a significant part of the selection process (e.g the military) and should be considered. It doesn't mean they won't let you in, but it may mean they would like to investigate further. If you are accepted you may not get the type of job within that profession you want.

Having said all that my advice is to avoid a career that is likely to worsen your anxiety. This may be stating the obvious but if you have a diagnosed anxiety disorder I would have thought the very last thing you should consider is a career that is known to be associated with stress. You mention the police and I would consider this a prime example of a career to avoid - even assuming it is an option.