Connecting flights Timing through Barcelona to Seville - Do I have enough time?

Check with United. You may be able to check your luggage through to Sevilla and get both boarding passes in the US, even though it's a different airline and separate ticket.

As I mentioned in your other post, 2 hours would be unreasonably close. 4h45m should certainly be safe.

I’d still call United and ask about changing to ticket to connect into Sevilla.

If we land in Terminal 1 at 9:00am from our United flight how long will it take to go through customs and pickup our bags? I found another flight at 1:45 from Barcelona to Seville on Vueling airlines............. Can we do that in 2 hours? It's all in Terminal 1.

The way I learned math, 9AM to 1:45PM is more than 2 hrs

Are you not able to do a single booking from USA to Seville, rather than cobble together 2 separate flights and 'hope' it works?

Immigration can take more time than customs which usually involves walking through a green door marked nothing to declare. That's after you have found your bags.
Then you trek to the Vueling departure gate where you face a full security screening to board your second flight. I haven't flown with Vueling for awhile so I am not up-to-date but the general advice for any bargain airline is to read the website and follow their rules exactly. You may be required to check in on-line in advance, and the limits on bag sizes, carry-ons etc. could be different than United. Two hours should be enough but that time will probably be more stressful than if you were travelling on a single United booking for the whole trip.