Can any company purchase emission allowances under the EU Emission Trading System and then not end up using them? My reasoning behind it would be to keep on increasing this commodity's price in order to make it more difficult for other companies to buy it. The goal would be to force others to become more sustainable businesses.
Yes, this is possible. Emission allowances under the Emission Trading Scheme can be cancelled, and some organizations have done what you suggest in your question. Most prominently, from 2008 to 2017, the non-profit Sandbag (now renamed Ember) ran a "Carbon Destruction" service, which allowed members of the public and businesses to put money towards buying emission allowances which were then cancelled, each removing an allowance of one tonne of carbon dioxide emissions from circulation.
This scheme, however, was discontinued in 2017 due to changes to the ETS:
What changed in 2017?
The EU ETS has long suffered from prices too low to drive emissions cuts – because of an enormous surplus of carbon permits. Following Sandbag’s research and campaigning, by 2017 it became clear that the system was going to be reformed. The first key change was the introduction of the Market Stability Reserve (MSR), which stores some of the surplus permits away from the market. The second key change was the cancellation amendment – ensuring that some of the surplus permits in the MSR will be permanently destroyed.
What effect does individual cancellation have now?
The European Commission calculates how much the MSR removes from the market each year. This calculation discounts any cancelled allowances. Now, the more Sandbag cancels, the fewer allowances the MSR would remove from the market. Therefore, following the reform, cancelling allowances will no longer lead to 1:1 emissions reductions.
So the scheme you describe remains possible, despite the 2017 changes, but is no longer as effective.