No, you do not need to report your purchase to either jurisdiction. Delaware will not require you to file an income tax return unless you are using the home for income-producing activities or are otherwise earning Delaware-sourced income. For federal purposes, you do not need to specifically report that you purchased a house, however you may deduct certain expenses, along with your main home such as property taxes (currently limited to $10,000).
Here is an article you may find helpful: How Does Your Vacation Home Affect Your Taxes?