Does ASAP Utilities work with web-based versions of Office/Excel?

No, web-based versions of Office/Excel do not support add-ins and macros.

Microsoft:

"Microsoft Excel desktop app is an extremely powerful tool used to manipulate, analyze, and present data. Sometimes though, despite the rich set of features, your organization might find it easier to use Visual Basic for Applications (VBA), a programming language, to create a macro that performs mundane, repetitive tasks or to perform some task that the user interface (UI) does not seem to address. Excel Web App does not support these features."
Source: http://technet.microsoft.com/en-us/library/excel-web-app-service-description.aspx#bkmk_VBAMacrosscripting (no longer available)

https://support.microsoft.com/en-us/office/differences-between-using-a-workbook-in-the-browser-and-in-excel-f0dc28ed-b85d-4e1d-be6d-5878005db3b6

Excel Online (formerly Excel Web App)

Update from Microsoft - December 2014:

Apps for Office

Excel Online only supports content apps for Office. Content apps integrate web-based features as content that can be shown inline with a document.

VBA and macro scripting

Microsoft Excel desktop app is an extremely powerful tool used to manipulate, analyze, and present data. Sometimes, despite the rich set of features, your organization might find it easier to use Visual Basic for Applications (VBA), a programming language, to create a macro that performs mundane, repetitive tasks or to perform some task that the user interface (UI) does not seem to address. You cannot create macros with VBA in Excel Online, but you can open and edit VBA-enabled spreadsheets without removing (or corrupting) the VBA contained in the file.

Source:
http://technet.microsoft.com/en-us/library/excel-online-service-description.aspx

Microsoft: Differences between using a workbook in the browser and in Excel