Lab coats, like other forms of PPE, are supposed to provide a final barrier of protection, for when all other risk mitigation strategies have failed. They are not just an expensive form of fancy cooking apron. If you are getting spills on your lab coat, you should take this as an opportunity to re-examine your laboratory techniques to try to establish why this is happening.
Contamination (including wiping hands) on lab coats should be treated at the time of incident, and contaminated lab coats should not be removed from the laboratory, unless in a contaminated laundry or waste container. Most workplaces will have a well defined plan for laundering lab coats, and strict guidelines for how often this should be done. You can search for just how common this is.
Good lab coats can be expensive, but think of them a little like fire extinguishers. Most days in the lab, you shouldn't even need one. However, you want to make sure that when you DO need them, they will work properly and do the job they are designed for. Regular laundering of a non-contaminated lab coat will keep it in top condition, but once contaminated with hazardous chemicals that cannot be safely neutralized, they should then be replaced.
So, the answer to your question is: no, you should not attempt to launder your lab coat. Buy a new one.
And a final 2 cents worth: Dirty lab coats are not a badge of honour; they are a sign of a sloppy chemist.