"Saddened to hear the news about Todd. Heidi and I will be keeping him and the entire Palin family in our thoughts and prayers,' Cruz tweeted.
Palin headed out to support Trump on the campaign trail on Saturday, and attended events on behalf of Trump in Arcadia and Plant City, Florida, on Sunday.
Snowmobiles are widely called snow machines in Alaska, and not to be confused with snow-making machines.
Mr Palin's accident happened on Sunday night, not long after his eldest daughter, Bristol, posted a picture on Instagram of her newborn daughter, Sailor, and brother Tripp with her father.
"Sailor's first time at Papa's cabin," Bristol captioned the picture.
Later that night, Bristol posted a snapshot of her 7-year-old son, Tripp, wearing a helmet, a jacket, and matching black gloves.
"And my future Iron Dog Boy #openface #handmedowns," she wrote, referencing the grueling 3219km off-road snowmobile race her father competed in last month.
It's unclear if Tripp was with his grandfather at the time of the snowmobile accident.
Mr Palin is a skilled Alaskan outdoorsman, and has won the the Tesoro Iron Dog - the world's longest snowmobile race - four times.
He was forced to scratch out of this year's competition after his partner suffered engine trouble.
While defending his title in 2008, Mr Palin was thrown 21m from his machine and broke his arm, but was still able to finish in fourth place.
- Daily Mail