How to I tell a Views Block to only pull Snippets belonging to the same OG as the container page?
As fullerja said, you can use a Contextual Filter but you may also need to define a Relationship first so that the view pulls in the OG that goes with the snippets. 

If this page is showing OG content, hopefully it has the OG's slug in the URL. If that's the case then you can use that path for the Contextual filter. 

These answers might be helpful for the specific configuration:


Organic Groups Views Contextual Filters
How do I display all the nodes in an organic group?