No. Tomba is a standalone tool Apollo export is one feature among many, and you can use everything else without an Apollo account at all.
An Apollo subscription only becomes relevant if you specifically want to pull leads out of Apollo.
What you can do without Apollo
Most of Tomba works with no third-party account:
Email Finder — find an address from a name and company domain
Domain Search — list every address at a company
LinkedIn Finder — find the email behind a LinkedIn profile
Email Verifier — check deliverability before you send
Bulk tasks, the browser extensions, spreadsheet add-ons, and the API
If you do want to export from Apollo
A paid Apollo subscription is recommended, not required.
The Tomba Chrome extension works on Apollo's free plan, but each search returns only the first 100 leads — usually too few for a real campaign. Apollo's free tier also limits some contact data: email visibility is reduced and phone numbers are unavailable.
A paid Apollo plan lifts the 100-lead cap and gives you the full Apollo database, which is normally the reason people export in the first place.
For the full walkthrough, see How to export Apollo leads using the Tomba Chrome extension.
You still need Tomba credits
Whichever Apollo plan you are on, exporting consumes Tomba credits, because Tomba verifies each email and finds phone numbers during the export. You need an active Tomba account with credits available.