Kaloi Akhilgov, a lawyer from the site, told the Reuters news agency: "As they drove he [Yevloyev] was shot in the temple ... They threw him out of the car near the hospital.
"He was discovered there and they quickly put him on the operating table, which is where he died."
Prosecutors have opened a preliminary investigation, Interfax reported.
The police have denied killing Yevloyev.
Yevloyev was a prominent opponent of Murat Zyazikov, the pro-Kremlin president of Ingushetia, a Russian republic in the northern Caucasus.
He is one of the most high-profile journalists to be killed in Russia since Anna Politkovskaya, an investigative reporter, was shot dead outside her Moscow apartment in 2006, provoking widespread condemnation of Russia's record on media freedom.
A posting on the website called on "all those who are not indifferent" to Yevloyev's killing to gather for a demonstration in Nazran, where Zyazikov's opponents have clashed with riot police in recent years.
The website is among the most-visited for news on Ingushetia and has openly criticised Zyazikov, who threatened to shut it down on several occasions.
Russian officials ordered the closure of the site in June, saying it was disseminating "extremist" views.
Moscow had also blocked access to the site late last year after it urged readers to protest against the local administration, which the opposition had accused of corruption and mismanagement.
Rosa Malsagova, the website's chief editor, announced plans to seek asylum in France earlier this month.