Bulk API will give you entity data and activity data (in ver 2). Entry into a campaign is not an activity the contact is doing, but rather it's something you are doing to them - so therefore it's not a valid activity to expect from Bulk API.
This information is available via REST:
This is used in the operational report "Campaign Entry Report" under the campaign asset options menu. Like all REST calls, max count is 1000. This would be 1 API call per campaign, per 1000 results. Time ranges are in Unix Epoch.