Is it possible to use Skyscanner's API directly via an AJAX call if my website is using HTTPS?

There are two ways to solve this problem:

1. Use CURL from your backend service. You can consider building your own backend API service that consumes Skyscanner's API.
2. Use a web server proxy. There is a comprehensive article from Digital Ocean about implementing a web server proxy in Apache. Your Apache config file will look similar to the following: 

<VirtualHost *:*>
    ProxyPreserveHost On

    # Servers to proxy the connection, or;
    # List of application servers:
    # Usage:
    # ProxyPass / http://[IP Addr.]:[port]/
    # ProxyPassReverse / http://[IP Addr.]:[port]/
    # Example: 
    ProxyPass /skyscannerapi http://partners.api.skyscanner.net/
    ProxyPassReverse /skyscannerapi http://partners.api.skyscanner.net/

    ServerName yourdomain.com
</VirtualHost>