Let me first point out that it is not the preposition that you should be primarily concerned about. Your sentence has other flaws - if this is your personal statement for university admission, you need to fix them. In this kind of situations, it is better to be bulletproof in your writing.
My experience [in/of/with] internship, work and social activities relating to business allowed me to became interested in business analytics.
The sentence is very vague - "experience in internship and work"? Internship is work too. If it is unpaid, then make that distinction - internship and paid work.
It is better to say what exactly it is that you have experience [in/with] - (a) specialized topic or field (financial accounting, marketing, taxes, business law, etc.), (b) tools (business software, database management, etc.), or (c) doing something (managing and organizing events/meetings, consulting clients, making coffee, etc.)
Also, a better alternative to "... and social activities relating to business" would be "business-related social activities". Again, in my opinion, this is a bit vague. What were these business-related social activities - potlucks? If so, then mention it. Were they advertising campaigns, charity events (CSR?), or awareness programs?
If you have already mentioned these in detail in previous paragraphs, then you don't really need to say "internship, work and social activities relating to business" - you can just say "my professional experiences allowed me ..."
A. Preposition: in
When we refer to a specialized sector, area, topic, or field of knowledge, we generally use "in"
Their experience in marketing and promotion has brought excellent results (Cambridge).
She has ten years’ experience in television and radio. (usually followed by a noun)(Cambridge).
... experience in fashion design, ... experience in data modelling, ... experience in financial accounting ...
This post Experience in/ with / no preposition in ELL by Yuri says
"Experience in" implies the person has been (professionally) trained in something (a special field of activity)...
B. Preposition: with
When we say we are good at using something (a tool), we often use "with". "With" is also used when the object is a noun.
I have 3 years' experience working with Adobe Photoshop.
I have 4 years' experience working with children.
... experience with Java, ... experience working with a wide range of statistical software, ... experience with computers.
This post Experience/Experienced: “With” or “In”? in EL&U says
I have experience / I am experienced + with + general noun (cars, animals, etc.)
This post “experience in” VS “experience with” in ELL has a great answer by Astralbee on the use of "in" and "with": A farmer works in a field with a plough.
C. Preposition: of
"Of" is often followed by situations or specific types of occurrence - Free dictionary forum.
After 'of', we mainly use gerund unless the word 'experience' is preceded by get/gain - by Yuri in Experience in/ with / no preposition
Do you have any experience of working with heavy machinery (Cambridge)?
Do you have any previous experience of working with children? (usually followed by the -ing form of a verb)(Cambridge).
The use of the prepositions is not enforced strictly, I think. We mix it up often when we speak. Then there is the matter of how they may vary between AmE and BrE. There is a comment by JavaLatte in Yuri's post: "According to NGram, there is a significant difference in usage of "have experience X" where X is a preposition, between BrE (of >> in > with) and Ame (in > with >> of)."
All three - "in", "of", or "with" - can be used with "experience". However, all three prepositions cannot be used interchangeably in your context without changing the wording of the sentence.
As I said, the use of "internship, work..." is very vague to me. "Internship" and "work" are not fields, and I don't think we can use "in".
My experience of a 16-month internship program allowed me to ...
My experience of working with the Finance team is what made me interested in ...
My experience in project management (gained during my internship or work term) sparked my interest ...
