Company X who is registered in Poland as a Travel Agent provides
activities (games, sports etc) for foreign clients visiting Poland by
subcontracting suppliers in Poland. Company X then invoices the
clients.

Company X is unclear how to calculate VAT on the invoice for clients,
the clients are all tourists but it is unclear whether the activities
can be classed as a tourism service or not in Polish law?

We understand there are two possible ways of calculating the VAT:

1. COMPANY X adds vat on their margin at 22% and does not reclaim any
vat included in the buy price, which could be at 0, 7 or 22%.

2. COMPANY X adds 22% to the to sell price and reclaims the VAT
included in the buy price which could be at 0, 7 or 22%.

Which is correct? Or is there another way?

(The buy price is the price X pays supplier and the sell price is what
the client is charged)

The next question is, if an offshore company supplies the clients to
us then does company X still charge VAT in the invoice they issue to
the offshore company (Gibraltar)? (Company X buys the activity from a
local supplier and then sells it on to the offshore company for a
profit).

I think the answer is clear-cut, your alternative no. 2 is correct.

Your final question, it seems, would fall under the 0% VAT as an
"Intra-Community supply of goods", but I have some doubts about this,
as you and the local supplier of the touristic activity remain in
Poland, i.e., not a simple export of goods.  Even if the Gibraltar
entity is selling the activity, there is an actual or implicit
contract for the service/goods to be provided in Poland.  As you say,
"the offshore company supplies the clients to us".

As a tourist, I have seen this documented with vouchers issued by the
offshore (Gibraltar) entity for the services to be provided locally
(in Poland).
I have no idea whether this falls under VAT  - or maybe escapes the
tax authority's notice.

Please note that this is a free comment and NOT an official answer to
your question, which only a Google Answers Researcher with a blue user
name can post.
And, of course, anything posted here is not legal or professional
advice, as you can read in the disclaimer below.