When Nevada held its first Republican caucuses in 2008, Philip A. Kantor walked to the library next door to his synagogue. He took in the spectacle, watching others ballot for their chosen candidate. But he could not take part — as an Orthodox Jew, he was forbidden from writing on the Sabbath. Frustrated, he promised himself to prevent a similar setup in the next election. But by the time state party officials announced the date of this year’s caucuses, Mr. Kantor realized that he would be barred again, unless they made special provisions. Mr. Kantor placed several calls to friends and colleagues he knew were influential in G.O.P. circles, including to his longtime friend Sheldon Adelson, a Jewish philanthropist and Republican donor. Within weeks, Mr. Kantor had a meeting with the Clark County party chairman. After considerable back and forth, Mr. Kantor was assured that he and other Orthodox Jews would be welcome during a special caucus Saturday night, hours after others would end.