The PHP 5.3 limitation doesn't really have anything to do with SendGrid's API, it's just our official library only supports 5.3 and above. You could easily swap it out for any generic SMTP library.
For example, Swift Mailer supports PHP 5.2 and up.
And yes, the web API will be sufficient for your use case. It's identical to the SMTP API, except that it's over HTTP.