The strikes hit the villages of Dhahyan, Baqim and Jawf on Thursday afternoon in northern Yemens Saada province, said MSF spokeswoman Malak Shaher.
The area is a stronghold of Iran-backed Shiite Huthi rebels.
The ambulance driver, who worked for MSF and the Jamhouri public hospital in Saada, was hit as he arrived at the Dhahyan bombing site, said MSF.
"The ambulance was hit as it arrived at the site of an earlier bombing. When people gathered to assist the victims, the same site was hit again. The driver and the ambulance were then hit in a third strike," it said in a statement.
MSF said the total number of wounded and killed was still uncertain, but that it had received 40 casualties, six of whom had died.
"This latest loss of a colleague is devastating, and it demonstrates the ruthlessness with which healthcare is coming under attack in Yemen," said Teresa Sancristoval, emergency coordinator at MSF.
"People there are being subjected to this kind of violence on a daily basis. No one, not even healthcare workers, are being spared," she said in the statement.
Dhahyan is located about 20 kilometres (12 miles) from the city of Saada, not far from an MSF-supported hospital hit in a January 10 strike that killed six people, according to an updated MSF toll.
MSF in December accused the coalition of bombing its clinic in Taez, southwest Yemen, wounding nine people including two staff members.
The coalition said it would investigate that claim although it has repeatedly insisted it does not target civilians.
More than 5,800 people have been killed in Yemen since the start of the Saudi-led bombing campaign against rebels in March, about half of them civilians, according to the United Nations.
The coalition launched its campaign in support of Yemeni President, Abedrabbo Mansour Hadi, after the rebels overran most of Yemen and advanced on his southern refuge in Aden, forcing him to flee to Riyadh.