Combining Separate Tables With Multiple Values

Here's my issue.

I have 2 spreadsheets. One has names, phone numbers, and addresses. The second has names and another number(Let's call it ID Number). The problem is that the first one has over 14000 entries, while the 2nd one has around 600. All the names in the 2nd table are also in the first one, and so I want to add all the ID numbers to all the matching entries in the 1st table, based on name.

I've been trying to do this with VLOOKUP, and I can't seem to do it, and feel like an idiot. Can anyone help me out?