You can easily embed an Instagram post into your page by using the Script Editor Web Part. The embedded post will look like this:

Step 1. Retrieving Embed Code from Instagram

Please open the Instagram post you would like to embed. In the bottom right corner you will see an ellipsis. Please click it:

A pop-up window with a list of options will appear. Please select Embed:

In the next window, please press Copy Embed Code:

Step 2. Inserting a Web Part

Please open the page you would like to embed the Instagram post into in edit mode. On the top of the page, please select the Insert tab. Choose Web Part. In the list below, please choose Media and Content, and in the next list, please choose Script Editor. Press Add.

Step 3. Editing Web Part

After you have inserted the Script Editor into your page, please click the arrow icon on the right of it. In the drop-down list, please choose Edit Web Part:

After the page is refreshed, you will see a button Edit Snippet. Please click it:

Paste the copied embed code into the field and press Insert:

Save the page. 

That's it!

Thank you for your attention.