Is there a way to schedule a scan of the contents of a website?

Just a minor question. Why would you want too? It scans them when you download the stuff from the website.

My wife has a small business website. We want to be able to scan her website regularly to check for any infections of the code. Recently someone was able to change the index page html and insert a trojan. I don't know how is was done as the site uploads are password protected. It seems to me that there should be a way to scan the entire contents of a website just like you can scan any drive. I have googled this topic but have not found anything.

you should better install and run an anti-virus software on the server that hosts the website. it's much more efficient.
if the website is hosted by a hosting company - ask them if any antivirus option is available at their site or if it possible to establish and run your own software on the host. In the latter case, you'll be able to purchase and run Kaspersky AV at the hosted site (this may be quite difficult though for a non-IT person...)

as an option, establish regular mirroring the whole contents of the website including all descendent files and subfolders (using wget utility for example or any other batch download tool that suites you best) - then, the AV running on the host that performs mirroring (on your machine) will check the mirrored contents...