Well, either you use some existing web software and ordering system, or you build your own. There are zillion desktop development systems from VB to FoxPro to Access, and the same holds true for web development, you have to go out and pick a language and platform for development. However, just like you can purchase QuickBooks for about $99, it would make no sense to build your own accounting system at a cost of 1 million or more that you can get for $99. The same goes for a web based ordering system, you going to purchase or use an existing shopping Cart. In fact, recommend j-street and a product called CARTGENIE since the desktop portion is built in access, but the back end part is web based.
Since you are talking about a web based solution, then your current software is of really no use here. You could certainly use Access to pull data from the web site (not at your work location that as you mention has no internet).
Any web application will use a database server for the data store. In fact a big portion of web hosting providers allow you to connect to the web site, and you could do so with access. However, in your case, you don't have internet, so it don't matter.
To send this order information to the location without internet. Just download the order information from the web site into an Excel Sheet, or in fact run the report from the web site and print it out, and then Fax it to the location that does not have any internet. I suppose you could also consider placing the data on a jump drive.
I mean since you don't have any internet connection at this work location, it don't matter much if you snail mail a floppy disk or jump drive as Excel or a table in access. For a few orders a day, the labor savings of importing vs. that of a weekly fax sent and data entry done on site will not likely even save the cost of building an export + import system into your software. I guess this really depends on the volume of orders here, but if you do the data entry over time (weekly), then you not have a huge task come the seasonal work time when orders have to be sent out.
At the end of the day, your ordering software, you're pricing, you're adding of products and inventory etc. will NOT occur on the software that runs without internet at the work location.
So, the important business rules of updating pricing, processing of orders now has to occur on the web site, and will not occur in the Access application anymore. If you had internet then you could most certainly have Access connect to the database server on the web site to pull down orders. In your case, just printing out the orders and faxing them to the location that has no internet is likely the most practical solution.
You really can't use your existing ordering system since it can't control inventory, and can't control pricing and can't know what has been purchased like the web site can. So, the web system is where the business rules and data must exist. Everything else becomes just a consume of that web data. So, a printout + fax, or an Excel sheet on a jump drive delivered to that location is what you are looking at.
I suppose you could build some type of import system into your existing Access application that could import the Excel or csv data that the web shopping cart produces. So, you need to find a shopping cart system. Most if no all have some type of ability to export order data to Excel or downlaod sometype of csv file.
If you have a huge price + product list then you could/would look for a shopping cart system that accepts some type of price and product list that can be uploaded. However, if it not a huge product list, them a onetime simple seasonal data entry session again makes so much more sense then spending large amounts of time to look for uploading of price lists when they are not that large and only done 1 or 2 times a year.
It is unlikely you are going to build your own web based shopping cart anymore then it make sense to build an Accounting package when one like QuickBooks can be had for about $99. Check out a few shopping cart systems, and find one that can download the orders file, you then just print that out, or as mentioned consider building some importing routines into your current system.