You seem to be exceeding your request quota. Your site’s errors say:
Failed to load resource: the server responded with a status of 429 (Too Many Requests)
I have very rarely seen users hit their daily limit (maybe once or twice so far). The limits are set at 2 levels: 10000 API calls, and 75000 base URL calls. Generally, 1 gallery = 1 API call. Each time your page reloads, it is a new call. Since Google does not support caching its results, if you have 10 galleries on a page and that page is loaded 100 times a day, you will get to 1000 API calls.
Now, if each gallery has 50 photos, there are two potential “base URL” calls involved for each, or 100 in total. The first is when the thumbnail is shown. The second is incurred only if someone clicks on the thumbnail to see the lightbox, or gets to the photo while clicking on the arrows in the lightbox. So, for each page load, you may expect a maximum of 50 photos x 10 galleries x 2 calls = 1000 base URL calls. If that page is loaded 75 times and each visitor repeats the pattern, you will hit your quota. Hitting either the API quota or the base URL quota on a given day will result in the error. You might need to figure out your usage pattern.
I also noticed that you seem to be using Jetpack’s image CDN, which is further storing a copy of Flickr’s photos on WP’s own servers. This is counterintuitive, since Flickr’s CDNs are well-optimized themselves, and you are reducing efficiency by leaving the Jetpack CDN on. You might want to turn that off.