Trinket.io founder here. We embed trinkets in our WordPress blog (blog.trinket.io) all the time, which is hosted by WPEngine (wpengine.com). One solution is to use a hosting provider like WPEngine that works out of the box. I have to imagine that many of the excellent independent hosing providers out there support iframe embeds out of the box.
If you’re rolling your own wordpress instance, the open source WordPress.org code allows for a domain whitelist of accepted automatic embeds. More on modifying that here. It depends on whether the site supports oEmbed or not. Trinket.io currently does not, so follow the non-embed instructions.
If you’d like to explore the use of plugins, I haven’t extensively researched these options but there’s the iframe plugin (as mentioned by Henrik Sommerland) and Embedly (wordpress.org/plugins/embedly/). We’re not an officially listed choice in Embedly but there’s a built-in tool for embedding any URL- just put the trinket.io/embed/blah/blah URL in there and you’ll be golden.
Sorry for the trouble! These restrictions are put in place for security reasons so there’s not much we can do to make your experience embedding Trinket.io content easier aside from suggesting a hosting platform with more lenient/customizable plugin or whitelisting policies.