I would suggest taking a look at Nutcache. http://www.nutcache.com/
The free version does everything a freelancer would need. You can create your invoices, email them to customers, see when the customer views it etc.
You can then enable payment methods directly in nutcache so that your invoice would have a link where the customer can pay the invoice. You would get notified of the payments. You can even have Nutcache send reminders as invoices get older.
I don’t know what you mean by making this integrated in your website. Usually an invoice is just that, an invoice like a PDF that you send to customers.
If you integrate this into your site, this means you will likely need your customers to have logins to view their particular invoices. This means you have to make all your users sign up, or you assign them accounts so they can pay your bills. This is way to cumbersome for people!
Imagine if you hired a plumber and then when they were done, they said you had to now go to their website, create and account, then pay your bill there. How annoying!
I think 3rd party invoice tools are just great, and makes it easy on people to pay the bill. If they send you a check or use cash, you can just pay the invoice manually in Nutcache and send them the paid invoice just the same as a receipt.
If all you want is a payment system on your website, so people can click a button to send money, they just use simple Paypal buttons right? Have them type their name and then type an amount. When payments come in, you can edit your invoices however you want to.
If not Paypal, you can integrate similar buttons from Dwolla, Webmoney, or Stripe, etc. These are just for accepting payments, not really full invoicing systems.
On my website, I send clients the invoice with Nutcache, it has a note and URL to pay at my website. When they go to my site, they click a pay button where they type in their name and an amount. I just apply the amount to the Nutcache invoice myself. As a freelancer, this is not too much work, since one person can only work with so many clients at a time, it’s not like I have dozens of payments flowing in to manage, so it’s no bother at all to do it this way.
If you have a Wordpress site, I would not lock your business’s invoicing system to some WP plugin. It would be much better in the long run if your company’s financial records (invoices, customers) are in a professional product like Freshbooks, Nutcache, Quickbooks or many others. Then if you can somehow integrate those to Wordpress, this would be better than storing all your records in WP through a plugin and then trying to move that data out to Quickbooks etc.
Good luck on your hunt!