Quick Answer:
The subject of "helping" is "exposure," or if you want the full noun phrase, the subject is "early exposure to pets carrying bacteria on their fur." The subject does not include "may encourage the proper..." because that is the beginning of the predicate (verb + object of the verb) of the sentence. The antecedent of "it" is "the immune system."
Long Answer:
This may be easier to explain using a simpler example to start:
Consider the following two sentences:
(1) The gymnast performed well.
(2) The gymnast impressed all of the judges.
These two sentences have the same subject: the gymnast. If we combine these two sentences into one sentence, we don't have to include the subject twice because that would sound repetitive:
(1) + (2) combined: The gymnast performed well and impressed all of the judges.
In this combined sentence, the gymnast did two things: She performed well, and she impressed all of the judges. The subject of the verb "performed" is "the gymnast," and since there is no other subject listed before the second verb, it is understood that the subject of the verb "impressed" is also "the gymnast."
This combined sentence above is grammatically correct, but there is another way of expressing the same idea. The second sentence can be expressed as a participle phrase that is added to the first sentence.
Note: A participle phrase is a present participle and its predicate. A present participle is the "-ing" form of a verb that is functioning as a verb. (If the -ing form of a verb is functioning as a noun, as in "Swimming is fun," then it is called a gerund instead of a present participle.)
If we combine the two sentences using a participle phrase, we omit the subject of the second sentence (omit second occurrence of "the gymnast") and convert the second predicate ("impressed all of the judges") to a participle phrase ("impressing all of the judges):
(1) The gymnast performed well
(2) The gymnast impressed all of the judges --> impressing all of the judges
(1) + (2) combined: The gymnast performed well, impressing all of the judges.
A present participle of a verb does not have a tense, so the sentence can be in the past tense, future tense, or any other tense, but the participle will always have the -ing form. (Side note: The infinitive is another verb form that has no tense.)
The participle phrase can appear before or after the subject of the sentence. It can even appear in the middle of the sentence. It is understood that the subject of the sentence also functions as the subject of any participle phrases added to that sentence, regardless of whether the participle phrase appears before or after the subject.
Examples of different locations of a participle phrase:
(1) Hoping he would not be recognized, Mr. Smith wore a disguise to the event.
(2) Mr. Smith, hoping he would not be recognized, wore a disguise to the event.
(3) Mr. Smith wore a disguise to the event, hoping he would not be recognized.
In all three examples, "Mr. Smith" is the subject of the sentence. Therefore, in all three sentences, "Mr. Smith" is the subject of "wore" as well as the subject of the participle phrase "hoping he would not be recognized."
Let's apply this to the OP's question:
OP's example: "Researchers suspect that early exposure to pets carrying bacteria on their fur may encourage the proper functioning of the immune system, helping it to attack only harmful things."
This sentence can be broken down into two sentences:
(1) Researchers suspect that early exposure to pets carrying bacteria on their fur may encourage the proper functioning of the immune system
(2) Researchers suspect that early exposure to pets carrying bacteria on their fur may help the immune system to attack only harmful things.
The author combined those two ideas into one sentence by converting the second sentence into a participle phrase: The subject can be omitted the second time and the verb is converted to its "-ing" form. Additionally, to avoid repetitiveness, the second occurrence of "the immune system" is replaced with the pronoun "it":
(1) + (2) combined: Researchers suspect that early exposure to pets carrying bacteria on their fur may encourage the proper functioning of the immune system, helping it to attack only harmful things.
Remember that the subject of the main verb in a sentence also functions as the subject of a participle phrase in that sentence. So the subject of "helping" is "exposure." The antecedent of "it" is "the immune system," since early exposure to pets helps the immune system to attack only harmful things.
