Can a small business build an internal investment fund?

Yes, but why? If you're a small business owner you probably have some kind of IRR that is decent (20-30% say), whereas even professional fund managers will struggle with 6-10% returns (depends on strategy otc). 
 

A small business doesn't "pool" funds - it just opens a brokerage account that is owned by the business. This happens all the time and is typically managed by a firms internal treasury team (mainly comprised of running bond ladders to match cash flow expenditures or future projects). A business that has future use of capital typically does not have the time horizon for equity or alternative investments unless this is for a purpose similar to a pension. There are some tax benefits that a business can utilize to make pension savings more affordable on the business side vs personal.

It sounds a little WeWork-ish to me. A few years ago on WSO we were discussing WeWork (while it was still the beloved unicorn of Wall Street). I, being right about 99% of things and dayum humble about it, too, was pointing out that WeWork's foray into VC, residential real estate, wave pools, etc. was an indication that something was amiss with the company's underlying business model. Just sayin'...