That would (I believe) require a pretty good understanding of PHP/MySql code, along with the requirements for eCart in order to insert the correct recordsets and queries for each page that eCart generates.
At the very least, you would need to be able to configure a connection to a MySql database on your remote server in order to use the recordsets (as you wouldn't be able to do it on your local machine) with the ability to log into it remotely. Not all hosting providers allow this, although some will configure access for your PC's IP address if you contact and ask them.
HOWEVER - I strongly advise setting up a testing server on your machine, for both this and future use. It would be massively easier to do than create a fully functioning eCart site by hand coding anything.
If you are on a Windows PC, I recommend XAMPP for ease of installation, if you are on a MAC, then MAMP seems to be the one to use.
Check out some XAMPP installation instructions HERE
Setting up a Testing Server for Windows
Setting up a Testing Server for Mac OS X
Just my tanner's worth.