Can I fill in the blank on 1099's on my computer and print it out instead of having to type it on my typewriter?

The only reason I can possibly imagine anyone would need to be filling out a 1099 is if they're a company.

I don't know if there's an official fillable form for 1099's as I've never done enough of them to make it work it, really. But whatever is available, you can call and ask what it is:
1-800-829-3676. (That's the IRS Forms number.)

I used to do W-2's in Peachtree, and most companies are going to use software to do stuff like this.

You can use software like W-2 Mate to print W-2s.