If I am purchasing an undervalued property, can I get an immediate HELOC after sale to finish some needed updates?

How much equity will you have in the property, @Tony Lake ? Have you talked with any lenders about this yet? What was their reaction?

Depends on the deal and what it appraisers for! Go get it tiger! Some lenders require a seasoning period 6-12 months. 

Based on comps that a RE agent helped with, we would be purchasing our family homestead and I would probably be walking in with $50-75k in actual equity. The home is rural, so would need a septic upgrade and some cosmetic updates to make it fully functional. 

I also wanted to have the line of credit open to help fund a brrrr that my wife and I are currently watching for. We currently own a small MH park that cashflows a little.

As long as this purchase is your primary residence, you can get a HELOC but only if you have enough equity. Based on your numbers, it does not sound like you do. Speak to your local bank, credit union, or lender doing your first mortgage. They will tell you if you have enough equity to get a HELOC or not. Nobody here can answer that question.