Yes. Both online shops with Big Cartel addresses (e.g., https://bigcartel.yourstore.com) and those with custom domains built on the platform (e.g., https://yourstore.com) come with full SSL encryption. If you’ve added unsecured assets (CSS, JavaScript, custom images) to your shop which are served over http://, they may not be covered by SSL, though.
In such cases, all you need to do is secure your assets and contact customer support to request an opt-in to a site-wide SSL. But you can be assured that checkout is always SSL-secured.