UK companies should sell you products VAT free, IF they are exporting the goods, AND, you have supplied a valid VAT number on company headed paper. (You are purchasing these goods for business purposes aren't you?)
In my experience it's no problem. I simply supply my VAT number by email, and that works. I don't have company headed paper. Some companies have refused, or found it so difficult to comply that they may as well have refused,.. in those cases I don't order from them. (In some cases it may be cheaper to pay UK VAT than to order in Ireland VAT free)
You have paid UK VAT, not Irish VAT, and as such it cannot be reclaimed from Irish Revenue. You may be able to reclaim it from the UK Revenue, but you would have to register with them, and that may be difficult, or impossible.
In the future you should have your goods delivered VAT free, that's the system... not paying VAT to foreign governments and then attempting to reclaim it.
edited to add: All is not lost... if the company is still talking to you they may refund the VAT,.. they would likely be entitled to do so, as you had supplied the VAT number prior to ordering.. they should have EXPORTED the goods VAT free in the first place. (Collecting goods from a supplier is not 'exporting')