I'm working in a SharePoint 365 environment and created a web part page through SharePoint Designer 2013. When I edit the page online, the INSERT tab has options for Text, Image, App Part, and Web Part. Image, App Part, and Web Part are working properly, but the Text option is greyed out. Is there something I need to do to enable the addition of text to my page?
You can't exactly add text directly to a web part page. You need to add a web part like the Content Editor Web Part, and then you can add text to it. If you click on a web part zone, the insert text button should become active, but if you click it, it will just add a Content Editor Web Part for you. Otherwise, you can just choose web part from the insert tab a choose a Content Editor Web Part (under the Media and Content folder). Then you can insert text into the content editor (by clicking the link in the web part that says click here to insert content).
If you want to add text to a page without inserting a web part, you need to create a Wiki page. Then in edit mode you can select any rich text area and the ribbon will display rich text editing controls. Then you can type your text and save.