First of all, you're translating 取材 as "interview" in your question, but it's actually a much broader word than that. It refers to the whole process of gathering information for a report or article, so it basically encompasses everything the writer describes doing in the rest of the text - interviewing and interviewing hundreds of actual parents, talking to shop assistants and academics, visiting a nursery school where piggybacks are still used, and even conducting experiments by attaching a camera to children. There's no one English word that perfectly matches the sense of 取材, but here I think "investigation" would be a decent translation.
As for how 取材 connects to the rest of the sentence, let's look at the phrase そんな疑問を胸に取材を進める. This is made up of the main clause 取材を進める and an adverbal clause 疑問を胸に describing it. The verb of the main clause is 進める, which means to move something forward or progress it. We can phrase the same thing quite similarly in English, translating 取材を進める as "move forward with" or "proceed with" the investigation.
Then there's the descriptive そんな疑問を胸に - this refers back to the question in the previous sentence, "What happened to piggybacks?" そんな疑問 could be translated as "such questions" in some contexts, but really I don't think there are any other questions implied here, so it's better to treat it as just "a question like that", which is just a fancy way of saying "that question". The rest of the clause is the basic XをYに structure, meaning "with X as/in Y", so in this case "with that question in my heart".
Put the whole thing together, and we get "With that question in my heart, I proceed with my investigation" for そんな疑問を胸に取材を進める. This seems like a pretty reasonable translation, though it might sound a little neater in English to just say you "carry out" the investigation rather than "proceed with" it.
Now all that remains is to connect it to the latter half of the sentence. Your existing translation of the rest of the sentence, "I came to know the charm of piggy-back-riding once again", seems perfectly good already, but you seem a little confused about how the two parts are temporally connected.
The simple answer is - the whole sentence is in the past tense, because at the time of writing the author has already completed the investigation and come to appreciate the charms of piggy-back riding. What might be throwing you off is that そんな疑問を胸に取材を進める appears to be in the present tense, but the present tense is very neutral and flexible in Japanese, and the tense of a sentence's main clause can often be considered to extend to any subclauses even if they appear to be in the present tense.
In this case in particular, the "present-tense" そんな疑問を胸に取材を進める is linked to the past-tense main clause by と, and と-clauses are always in the plain present tense - 取材を進めたと… would be outright ungrammatical. We can tell that this is a past action, however, because と implies something that came before whatever follows, so logically speaking it must be further in the past than the past-tense main clause.
So to put the whole thing together, I'd translate this sentence as something along the lines of:
"As I carried out my investigation with this question in my heart, I came to appreciate the charm of piggybacks all over again."
