Yes you can set up an account online. Yes you can tie your bank account to it so you can transfer funds electronically.
Their user website is pretty good. Not only can you purchase or transfer funds, or reallocate, you can set up recurring transfers, you can research funds, and you can get personalized breakdowns of your account performance over time by a variety of metrics. All from the comfort of your home internet connection.
Just to be safe, since you're dealing with your life's savings, I'd probably keep an updated OS and antivirus like MSE running on a hard wired connection. No wifi, no public computer access. The chances of having your data intercepted are small, but it is super easy for someone to do if they know there's a cool million or two waiting for them.