Short answer: no, you can’t stay airside all the way from Terminal 4 to Terminal 7 right now.
Terminal 4 (American) is connected airside to Terminal 5, 6, 7, and 8, but the connection starts from Terminal 5, not directly from 4. Because of the construction and how the corridors are set up, you usually can’t walk from Delta’s terminals through T4 and then continue to T7 without exiting security.
What most people end up doing:
Option 1 (most common)
Exit security at Terminal 4
Walk outside on the sidewalk to Terminal 7 (about 10–12 minutes)
Option 2
Take the LAX landside shuttle between terminals
Then go through TSA again at Terminal 7
If your goal is just meeting family arriving on United, the easiest thing is usually:
Wait outside the baggage claim at Terminal 7
LAX is improving airside connections, but Delta terminals (2/3) are still separated from the T4–T8 corridor, which is why this gets confusing.