Is there a way to edit the fields

Is there a way to edit the fields? Specifically, I would like to collect Company Name and Phone Number as well

Adding fields doesn't seem to be supported yet -> so, is there a way to edit some source files to include the new fields and have that data sent in the email?

Hi,
this is very simple contact plugin,
I suppose you need more features, don't waste your time for customization when you can choose from more than 15 different solutions.

Two main options:
1) Use Contact form - module /also from component package/
2) Use Form creator component which allows you to build any form you want

Check JED (Joomla Extensions Directory) resources or use Joomla app store from Joomla itself.

p.s.
Unfortunately, because you are using Joomla 3.x - probably you can not install free : aiContactSafe - which had all features you wanted and much more.

But you can try to use : BreezingForms or JSN UniForm - of course PRO versions always have more features, but also Basic/Free should be useful.