If the property owner gives the HOA legal permission to enter his or her property then the HOA has the right to do so. Absent that permission, the HOA should not enter the property. In the scenario set forth above, I suspect the HOA did not want to do so for fear of being accused later of some wrongdoing. A loud alarm sounding for 48 hours rises to the level of a general nuisance which should have been dealt with by the association either by obtaining the owner's permission and entering the home or by contacting the alarm company to deactivate the alarm remotely.
Regardless of whether the HOA can go into the property, you are probably not going to be able to force them to do so. HOAs have discretion in how they enforce covenants and govern the association. An HOA is well within its rights to refuse to enter someone's home. This is certainly a frustrating situation and there may not be anything you can do to address the immediate problem of the running alarm. Try calling the city/county often enough and they may eventually find it to be an emergency. Good luck to you.
I hope that this answer helps you in a general sense, but it does not constitute legal advice, nor tax advice of any kind, and does not create an attorney-client relationship between you and myself or my firm.