Which year can I start to apply graduate program in United States? As the question stated, is it possible for me to start applying graduate programs during forth year of undergraduate studies?
What matters is whether you will have completed your undergraduate degree before the graduate program starts.

For example, suppose there is a program with application deadline December 2017, with enrollment in September 2018.  You can apply this year as long as you expect to finish your undergraduate degree before September 2018.  So if this is your last year and you expect to graduate in spring 2018, you can apply this year.

For students on a traditional US schedule, this would usually be the fourth year of their undergraduate studies.  But some students might be finishing their undergraduate degree in a shorter or longer period of time.  E.g. if this is your fourth year as an undergrad, but you expect to need one more year to finish and will graduate in spring 2019, then do not apply to graduate programs this year.