Tl;dr: Yes, you are required to contact all airports and heliports within a 5-mile radius.
Unless you have a part 107 certificate you are operating under the rules of 14 CFR Part 101, Subpart E - Special Rule for Model Aircraft. Since the FAA is generally prohibited from setting rules for hobbyist operations, the only specific regulations are under this subpart. They are frustratingly unclear and refer most rulemaking to undefined “nationwide community-based organizations.” Although which organizations, or even what such an organization consists of, are not defined, the FAA works closely with the Academy of Model Aeronautics which publishes a handbook for safe operation.
One of the few specific regulations subpart E does include is the following.
14 CFR § 101.41 (e) When flown within 5 miles of an airport, the operator of the aircraft provides the airport operator and the airport air traffic control tower (when an air traffic facility is located at the airport) with prior notice of the operation.
This means that to be in compliance you must notify all 32 heliports in your area. Note that it does not say you must get permission, only that you must provide prior notice. It also doesn’t say what kind of notice, nor how far in advance, nor if you must notify them every time you fly or if it can be an ongoing situation.
Mongo’s suggestion of mailing a letter to each one would be a good approach. Make sure you keep a copy of the letter sent and a record of who you sent them to. That way if a problem arises you have documentation that you did make the required notification.
Beyond notification, there is no mention of airspace classes, flight restrictions or anything else. It only specifies that:
14 CFR § 101.41 (d) The aircraft is operated in a manner that does not interfere with and gives way to any manned aircraft
That’s not to say that you don’t have to follow flight restrictions. That would fall under the elastic clause:
14 CFR § 101.43 Endangering the safety of the National Airspace System.
No person may operate model aircraft so as to endanger the safety of the national airspace system.
If you are planning to fly for research then you should certainly get a part 107 suas pilot certificate. Part 107 regulations are much more well-defined. If operating under part 107 there is no requirement to notify the airports and heliports. An suas license requires you to know how to read charts and know what type of airspace you are operating in. By default part 107 is restricted to uncontrolled airspace (class G). To operate in controlled airspace you must apply for a waiver and follow the rules of the waiver. Once you have a waiver for controlled airspace the requirements for getting permission depend on the specific airspace you need to fly in. Under part 107, in class G airspace there is no requirement to notify anyone.