Under what situations can Congress conduct investigations and delegate authority to administrative agencies?

According to the House of Representatives website, the Constitution does not outline specific powers of investigation for Congress, but instead notes:

“All legislative Powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives” (Article I, section 1).

In other words, Congress has the legislative power to pursue investigations as it sees fit. This is why there are congressional hearings for everything from jaywalking to opioid abuse to elections. Congress has the right to form committees, with investigations typically falling under the purview of a particular committee. The House Oversight Committee for example, which is a standing committee that has existed since the early 1800s, investigates how federal dollars are spent by recipients. Special committees may also be convened from time to time, such as the Special Committee to Investigate Pearl Harbor or the Special Committee to Investigate 9/11.
When it comes to administrative delegation, things become a little more complicated. The Supreme Court has wrestled with Congressional delegation powers for hundreds of years, with one of the more recent examples being the SCOTUS rulings on the Affordable Care Act, a.k.a Obamacare. The 2012 rulings determined that the individual mandate for healthcare was, in fact, constitutional due to Congress' power to make laws regarding taxation. Because Congress passed the ACA into law, which was within their purview, the SCOTUS affirmed Congress's power regarding taxation.
In short, when Congress feels out of its depth, that its resources will be insufficient, or that an issue has created a conflict within the legislative body, it has the power to delegate authority to outside agencies.