Is my mssql 2008 standard suitable for web edition 2014
Web edition looks like Standard edition except "It is only available through an SPLA" going by the specs listed in the article you linked to.

With regard to a 2Gb limit, are you thinking about Express edition (or MSDE as it was called pre 2005)? That has had a per database size limit of 10Gb since SQL2008R2 (it was 4Gb in 2008 and 2005 and 2Gb in 2000 &amp; SQL7). Note that is is per-database and data files only, so you could have several databases each under 10Gb that consume much more than 10Gb in total, and log files don't count towards the total. The key limit on Express is that it won't use more than 1Gb of RAM per instance which will limit performance for large data. You can mitigate this a little by running multiple instances. So running Express in a VM may be suitable for your needs and the least hassle option as you don;t have to worry about differences between SQLServer .

I don't have much experience with Azure SQL so I can't comment in any detail on the relative suitability of that for your needs. There are some differences in what is supported so you'll need to find documentation on that and test any potentially affected parts of your application. Be ware of which resources you use for this research: there have been many changes recently so articles form a couple of years ago may be pretty much irrelevant. See this SO question and other similar recent questions for potentially useful references. As well as feature compatibility you'll need to judge the performance requirements of your app against the "Database Throughput Units" (and therefore the cost) of the account types you are looking at, see this MS page for more details about the performance specs for each service tier.