Can a homeless person vote in a United States election, assuming they are a citizen? I suppose one barrier would be that they don't technically have a "residence", so it's not clear what their polling place would be, but I don't know if there are provisions for that (like is there a special ballot for homeless people that just has state or national level items?).
The states handle the voter registration process, so the requirements will vary from state to state.
In general, registrants will be asked for an address and place of residence - which will be a barrier to homeless voters.
The Veterans' Party has a table summarizing voting requirements related to homelessness. They cite the National Coalition for the Homeless as the original source, but the page appears to have been moved.
Additionally, a few states have published guides for homeless voters. Wisconsin is one such state (find their guide here). They note that a "residence" does not need to be a valid address; it can be a general description of an area where the person generally stays.
As stated here, disenfranchising the homeless is considered a violation of their rights under the Fourteenth Amendment's Equal Protection Clause. Specifically:
One of the first court cases regarding homeless voting was Pitts v. Black (1984) in the US District Court of Southern New York, which ruled that disenfranchising homeless citizens is a direct violation of the Equal Protection clause of the Fourteenth Amendment... The Election Board denied the individuals the right to vote because they resided on the street or in shelters. The Election Board contended that residency required some claim (such as rent) or ownership of the area on which they resided... The District Court defined the meaning of "residence" as any fixed location which the individual intends to inhabit regularly. This ruling encompassed all homeless, including those residing on streets and in parks. (1)(2)
Collier v. Menzel (1985): The Santa Barbara District Court case established that a residence could be a certain location rather than a specific address. Howard Menzel, the county clerk, rejected three voter registration applications on the grounds that proper addresses were not provided. The applications simply stated a public park as the applicants' residence. The court overruled the clerk's decision and defined a residence as any fixed location where a person habitually sleeps and where living quarters are set up. In their final decisions, the court stated that denying a citizen the right to vote due to residence in a public park, is a violation of the Fourteenth Amendment's Equal Protection clause. (2)(3)
Walters v. Weed (1988): The California Supreme Court judged a case concerning voter precincts of those who are between residences. The court ruled that when a person leaves his former place of residence and has not yet settled in another permanent living place, then the individual may vote in the precinct of his former residence. (4)
"Marthaann E. Pitts, et al., Plaintiffs, v. Robert S. Black, et al., Defendants No. 84 Civ. 5270 (MJL)". court case. United States District Court for the Southern District of New York. Archived from the original on 18 February 2009. Retrieved 4 December 2011.
"Court Decisions on Homeless People's Voting Rights". National Coalition for the Homeless. Retrieved 4 December 2011.
"Collier v. Menzel, 176 Cal. App. 3d 24 – Cal: Court of Appeal, 2nd Appellate Dist., 6th Div. 1985". court case. Court of Appeals of California, Second District, Division Six. Retrieved 4 December 2011.