I have a murray snowblower with electric starter. The starter is moving but not engaging with the blower. What can I do?

A little vague "the starter is moving" Does this mean the starter is making a spinning sound and the starter wheel won't engage the flywheel "the blower" Is your battery fully charged ? The little wheel slides on a shaft and the starter motor spins that shaft so it rises and catches your fly wheel to spin the motor to start.
Basic starter problems.
1. Not enough current for the stater to work and spin fast enough to engage flywheel or helper spring for starter wheel is worn out.
2. Ground for starter is bad and or battery cable to starter is loose or also bad connection
3. Starter teethed are worn and slips past the teeth of the flywheel.
4. The flywheel wheel teeth are also worn.
5. The starter bolts are loose or its out of alignment to mesh starter wheel to flywheel and shims fell out if used on installed starter.
6. The Bendix on starter is bad and the motor is pulling too much battery juice and makes battery seem low on charge.
7. shaft for the little wheel is gummed up or corded and the wheel won't rise on the shaft high enough to engage flywheel teeth.
8. Solenoid for starter is faulty or bad connections on solenoid
Listed are the basic problems you can have for starters Check the battery charge and all connections and go from there. If you feel you need to get a new starter take it off the old one lay it on the ground. Put foot on starter to hold it down. Ground the starter and tap the hot side of the correct volt and see if the starter spins and little wheel goes to little window in housing. If this works its only a partial test It can seem good but under load it could be week. This seems like a lot of explanations for what you asked but I tried to cover all areas. Good Luck and Happy Holiday. See what you come up with and post back.