Keep track of costs of supplies, number of hours you have in this project, and then when finished honestly tell us what kind of results you get. It's not practical, but it would be nice to know in 10 years what it looks like too. The reason I say all of this is because $150 is a very reasonable price to have a hide properly tanned.
If you insist on doing it in house, you will need to skin the coyote, then flesh every ounce of meat and fat from the skin. Leave NOTHING. After fleshed, turn the ears inside out, and remove the earbutt muscles. Let this hang fur side out overnight, then the following day shake all loose salt from the hide, and salt it again with fresh salt. This time let it hang fur side in until the hide is completely dry and hard. You will now need to re-hydrate the skin in a 10% salt brine solution. It could take some time for this step. You will now need to pickle the skin in a good pickling solution. The recipe can be obtained from the acid supplier, but it will basically consist of water, acid, bactericide, and salt. On a stinky animal like a coyote, you will need to add a good taxidermy deodorizer with this step as well. Once pickled, it's time to shave the skin. For a deer, you really need a shaving machine, but since you are just doing a coyote you can get by doing it by hand with a scalpel or skiving knife. After shaving, it is time to degrease. Use a quality degreaser intended for taxidermy use. I use tannery degreaser, but Knoblachs makes a good one too. I do this in my pickle mixture. You will need to agitate the skin A BUNCH. Tanneries have paddle vats that continually agitate skins while they degrease, so you will need to do this manually. It will take some serious elbow grease, but you can get it done. Once degreased, neutralize your skin in a sodium bicarbonate/water mixture until your skin has a pH of 4.0. Now spin out the excess water and dry the skin with a towel. (You don't want too much moisture in the skin) Now you can apply your tan. I use Liquatan, but McKenzie Tan or Trubond 1000 will work just as good. Roll the skin up hair side out and let it sit overnight. The next day, wash the skin in Kemal-4 until clean. Finally it is time to apply your tanning oil. Once oiled, let the skin dry hair side in, and while it dries you will need to stretch and break the skin several times a day until completely dry. Your final breaking can be done by dragging the tanned skin over a table edge, until the leather softens to your liking. Warning: While these are the proper steps for home tanning, the results will still not be the same as taking it to a taxidermist and letting them have it commercially tanned. Like I said before, I think the price of $150 is more than fair. I have added a list of supplies needed for the job, should you decide to do it yourself.
Skiving knife (pretty cheap, available at most taxidermy suppliers)
10lbs of salt
Tannery Degreaser
Taxidermy Deodorizer
Acid (Formic or Saftee)
Bacteriacide (I use NBU)
pH testing strips (0.0 - 7.0)
sodium bicarbonate (baking soda)
brush on tan of your choice
Kemal-4
tanning oil of your choice
Good luck and have fun. I think tanning is something everybody should try once. You will learn a lot and gain a better appreciation for what is involved in producing a quality tan, whether it be for taxidermy or just a wall hanger or throw. Let us know how your project goes.