Your first question
Is there a way to email the form to the vendor, have them fill it out electronically and then import the information to append the table?
Yes there is a way to send a message from access via email. Its not a form per say but you could send it and then extract the data. I don't recommend this method.
Second Question
Would it be better to create some sort of web-based portal for them, where they could query their vendor number for outstanding requests, fill them in and it would automatically append the table?
This is certainly the best option out there. However the knowledge required to do this is quite substantial. If you want we can discuss this in further detail, email me at jpgomez80 at gmail dot com and I can communicate more descriptive thoughts about this. But to get you started, Access is a database, any database can be taken to the web. Your choice of programming languages is vast. PHP, ASP, ASPX, JAVA and the list goes on. You will need to have a computer(Server) configured and setup to host such desired solution. This forum as well as serveral other ones go over asp, I suggest you go to
www.asp101.com and start with basic asp before moving onto ASPX. Again contact me at my email and I can explain this in more detail.
The last option you could do is send out a spreadsheet and then import the new data into your database (aka append). However the web based solution is easier to manage than this particular solution.