Is it permitted to go to the polls to vote before davening shachris? Is voting included in "chafatzechah" (your own needs) that Halacha forbids or is it a dvar reshus (a neutral act), a dvar mitzvah or none of the above?
Perhaps your question can be answered with the following.
It says in the Mishna Berura (89:17) that it is forbidden for one to begin any work, become involved in personal needs, or set out on a trip before praying Shacharit. The pasuk in Psalms is quoted (85:14) [In Kitzur Shulchan Aruch, 8:1-2] and used to explain that first the Righteousness of the Creator must be declared (by praying), and only after that, can one set out to fulfill their needs, desires, and personal endeavors.
If voting falls under "personal needs" then the above would lead one to believe that it is forbidden to vote before praying Shacharit. Furthermore, I cannot fine a source that classifies voting as a "dvar mitzvah" and even if it was, it would not be permitted as one must pray before fulfilling other mitzvot.
One may argue that a mitzvah that one will lose the opportunity to fulfill may take precedence over the obligation of the Shacharit Prayer. However, I would contend that since voting is something one can do the entire day (and in some states even via mail) it would not supersede the obligation to pray Shacharit.
I would say voting is Dvar Mitzvah because for every vote that goes towards a moderate party, the extremists are another step away from coming to power. (This would apply in most western voting systems, but I'm not sure about the US.)
Being a positive thing to do that is not for your own needs means that you could probably vote before praying if need be. However, the whole point of not doing things before tefillah is so that you don't get preoccupied and miss the tefillah. So if you are able to, it is commendable to pray ("daven" for our Yiddish-speaking brethren) first.