the cancellation of a contract can take place
a.- in a friendly way, through agreement of both parties.
b.- at the request of one of the parties to the contract under qualified circumstances ("unillateral cancellation"). Among these we can count the impossibility not due to negligence, the harship and the breach of essential contract obligations.
Considering the above the reply to your questions is:
Please could someone clarify and confirm who has to issue the 'Legal Notice of Contract Cancellation? You or your solicitor, if you want to cancel.
Can they refuse to allow us to cancel? Yes, in such case your only option is to sue the developer at court and your chances will depend on the proof of the "qualified circumstances" I mentioned above.
If they do allow us to cancel, does the notice still have to be issued, or is there another legal document to cover this, which prevents either party from pursuing the matter further ?
If the cancellation is agreed by both parties then an agreement is signed by both parties (we would be in case a as explained above) and the matter is settled at once, no strict cancellation notice is necessary then. However Developers take more seriously the cancellation issue when they have it in writing. Preliminary talk is a less aggresive approach, but it also might be effective.
We asked her to cancel the contract 2 weeks ago, and still nothing has happened - except that she has spoken to the builders and they have refused to refund us any of the 72,900 euros we have paid (44% of purchase price).
Your solicitor is surely trying to try a friendly cancellation as exposed above in case a, instead of unillateral cancellation as referred to above, case b. As far as I can tell, it is her the one who has to evaluate which options there are of getting a friendly agreement and under which conditions. I would also suggest you do not take strictly time issues or that you underestimate your solicitor's capacities. Please note that in the event the option a is not available, then your only option would be option b as explained above, which will imply a court proceeding to achieve the refund of the moneys paid. Court proceedings usually take years and involve additional expenses, therefore the decision should be taken carefully.
I would like to add that the plaintiff in a case carries the burden of proof, which means that you as plaintiff will have to give proofs of the "qualified circumstances" which justify the unillateral cancellation of the contract. And if you lost the case, you would probably have to pay the other party's court expenses. For a case where 30.000 € were at stake, such costs could be round 4.000 €.
Consequently, it would be wise that you checked with your present solicitor your chances to get the refund at court, or at least that you want her to send the cancellation notice, assuming all the risks. If you are not satisfied with her reply (this happens frequently, two solicitors do not currently see the case in the same way) then you can look for advise with another solicitor.