Can you get share count data for Facebook or Twitter?

Unfortunately not.

Previously, users could get share counts for both Twitter and Facebook, but both of these social networks shut down the APIs which offer this data.

You can read our update posts to find out more about the specifics at the time:

Is there anywhere I can get this data from?

Essentially, Twitter just completely closed their API down, then made the last 30 days data available through a (very expensive) paid service, but not the historical data. Buzzsumo cached a load of old data, and they do pay for the new data, so they probably have the most complete data set you can get for Twitter shares.

In terms of Facebook, there is no paid option, the API just depreciated and was replaced by an API that requires browser authentication. To some extent, they have an API you can 'hack' with proxies, but the data is really unreliable and incomplete - it isn't a good solution. Again, Buzzsumo is possibly your only option here, as they seem to be taking snapshots using browser instances - this means the data is not (and never will be) up-to-the-minute accurate.