My grandfather purchased some shares of a couple different companies and I would like to know how I to sell them? Or whom I could speak with to expedite the process?
Mid-Continent Oil and Refing Company Limited
Aluminum Top Shingle Corporation
There's really two questions here. The first is
who do I talk to?
It seems to me that the transfer agent should be the main person you need to get in contact with.
See this question and the answers for some advice. You will have to do some research yourself and/or hire a service to get more details on the company. Again, you are looking for the company's transfer agent.
how do I determine the value of the shares then sell them?
If the company still exists, you have to find someone to buy them. You can contact your broker and see if they can help you sell them on the open market.
If the company is liquidated, then the transfer agent should also be able to help you cash-out the shares. Although, there is a chance that you don't get anything if all that money has been used up.
Investopedia page for more info on paper shares. Google brings up many, many other sources of information as well.
Keep in mind that you may need to complete some legal stuff if you inherited these shares. An estate lawyer (if you don't already have one) can help with that.