Sorry to hear your Dad is not well.
I went through the process to get my permit in NY in Fulton County.
Start the process by going to your local Sheriff's office and getting the application package.
When you submit it you'll need to be fingerprinted, have a background check run, provide some passport photos, and the names of 3 or 4 character references that you have known at least six months, are not related to you, and are residents of your county (this requirement clearly shows the anti-immigrant roots of the permit system from the Sullivan Act).
They will send a questionnaire to your references and your application will not move forward until they all reply so choose people that will send it back in quickly. You'll have to attend a class at the Sheriff's office and they'll tell you when it is. Some counties and departments only do them once a month.
After they have all your paperwork, your reference responses, a clear FBI background check, and you've attended the training they'll contact you to tell you to schedule a meeting with the issuing Judge you have been assigned to. Getting a timely meeting with the Judge can be difficult. You'll sit down one on one with the Judge for an interview and you'll either walk out with your permit or he'll deny you.
There is an appeals process if you are denied but it's probably a waste of time and money since the other Judges are unlikely to overrule the one who denied you.
If you are a 48 year old business owner with a clean record I cannot see you being turned down but the process does take some time. The county you are in is the deciding factor on how long. Some get people through in a month while others take a year.
NY issues two types of pistol permits. One allows you have the gun in your home or place of business (this is either or but not both), at a range, and unloaded and locked on the way to and from the range. The other allows you to carry almost anywhere concealed. The issuing Judge can put restrictions on the carry permit to make it for hunting, hiking, fishing, target shooting, etc. but violation of these restrictions is not a crime.
From a legal point of view if you are stopped and are carrying you either have a permit or you don't. If you have violated your restrictions and word gets back to the Judge he can call you in and revoke your permit though. In most counties they will issue a permit with restrictions at first, unless you have a compelling reason to need your handgun daily due to a risky job or one where you carry large amounts of cash or other valuables, and after one year you can go back and see the Judge and have the restrictions removed. If you have restrictions the normal thing is to join a club with a range can use whenever you want and keep a shooting box in the trunk with eye and ear protection, ammo, and targets. This way you are always "on the way" to the range.
If you just want to be able to take possession of your Father's guns then you can go for the on-premise only permit which should be pretty easy for you get. If you have a business that requires you carry large amounts of cash or transport valuables, or be in remote and dangerous locations alone then you can probably get a concealed carry permit fairly easily as well given your age and status as a business owner. If you have a reference that is a friend of the Judge it will be a help too
.
Once you have a permit the make and serial number of all your handguns will be listed either on it or on a separate card that you must carry with you when you have one of them. If you buy more or sell any you have to go back each time and have it added or removed at the Sheriff's office. You'll have to have new ones added before you can take possession.
To get your Father's guns he'll have to ship them all overnight to a NY FFL. Since they are coming into NY for the first time they will have to be sent to Albany to be test fired so the round can be added to that monumental waste of money ballistic system NY built. I believe the cost is around $35 per gun.
Good luck. Let us know how you make out. I'll warn you that the process is long, frustrating, discriminatory in its design, and depending on your county can be very painful.
It would have been much easier if your father had collected rifles!
Edit: With the permit for target shooting at least in my county you could make stops on the way to and from the range if they were reasonable e.g. Wal-mart for some ammo and targets, the gas station for gas and coffee, or McDonald's for a burger, etc. Stopping off for a beer or to do your grocery shopping would be prohibited.