When my home was built in 2016 the HVAC runs from the furnace in the basement up through the house uses 7" R4.2 insulated flex duct. I am in the process of finishing my basement and I would like to tap into the existing runs. I am not sure what type of register to use to blow downward from the ceiling perimeter of the main room or how to accomplish this. Any help would be greatly appreciated.
I don't believe 7" or 8" ducts are trunks, and therefore you shouldn't be splitting them. Each vent requires its own branch duct from the trunk.
Trunk lines are almost always rigid rectangular duct of 10x12" or larger. You install a starting collar and connect your flex duct to that.
If your builder didn't properly facilitate new connections for your basement, you may need to have a tinbender come in and extend the trunk appropriately.
It is probably a bad idea to try to divert even a small amount of circulation from the existing furnace blower into the basement. It is highly likely that the overall system was designed and built to deal just with heating your upper floors. Home builders do not plunk extra capacity into something like a heating system because it adds to their cost and eats into profit margin.
Another consideration for heating your remodeled space in the basement would be to add an additional floor standing direct vent heater into the basement. When placed along an exterior basement wall that is not fully below grade the concentric direct venting can come through the upper part of the wall. Then all you have to deal with is to run gas and electric to the added heater. The new heater can warm up a pretty good sized area via its internal convection fan. Sometimes the direct vent can even go out through the rim joist above the foundation.
One of these heaters with an integrated blower is recommended because it can do a lot better job of circulating warm air to a larger area. Some models that do not have the integrated blower would be a lot less effective.