US-based monitors SITE Intelligence Group said the Islamic State (IS) jihadist group claimed the attack, identifying the target as a meeting of Huthi rebels inside the club in Radaa, a town in Baida province.
The Sunni extremist group oftens targets Shiites, branding them heretics.
A security official said: "Huthi fighters opened fire on the car before it attacked the club, causing it to explode at the gate, leading to the deaths and injuries."
A nearby school was also damaged, the source said.
Late last month, dozens of suspected Al-Qaeda militants, civilians and a US Navy Seal were killed in a US raid on a compound in the same province.
Al-Qaeda and its jihadist rivals of IS have exploited a power vacuum created by the conflict between the government and the Shiite rebels to expand their presence in Yemen.
More than 7,400 people have been killed in the war since a Saudi-led coalition started a military intervention in support of the government in March 2015, the UN says.
That toll includes around 1,400 children.