Invoice isn't required before billing a customer (very few SAAS products I use send one), but you would need to send your customers a receipt (aka confirmation of a transaction) for every renewal. I am not sure if you are legally required to do so, but it's such a common practice that your customers are very likely to expect it.
1. It's one of those things that's part of the user experience that people have come to expect.
2. Another reason is lowering chargebacks...
Many people signup for a trial / paid account and forget about it. Unless you send an invoice receipt via email every billing cycle, they'll end up realizing after a few months and then try to chargeback with their credit card company... and reducing your reputation score with your payment processor.
TL;DR You do indeed need to send receipts via email every billing cycle.
First, you should really be selling contract periods of 1, 2 and 3 years preferably with a minimum of quarterly in advance billing. The reason for this is that one day if you do decide to sell, the acquirer will discount your month-to-month subscribers, in essence lowering your perceived valuation. The other reason is it's very powerful on your cash flows.
You should be able to provide a record of the transaction on their credit card (preferably on demand if the user needs to see it electronically). This doesn't necessarily have to be done inside software. Depending on your size you can initially just have an administrative employee once a quarter run through credit cards via your credit card processing system, flagging any cards that were declined. This way you can save your product development efforts for features that count.
If you've sold to larger companies, you can expect that they may not do credit card and will force you to take checks. In this cae I would actively pursue EBT or anything you can do to stay away from the game of larger companies delaying your checks in accounts payable and abusing your payment terms. Always be willing to shut off customer's access if payment is late regardless of the mechanism of payment - and set that perception clearly during your sales process and reinforce it in your operations and in your contracts with those companies.
As for the legalities of receipts, you really should check with your accountant. And don't forget about potentially thorny tax issues when you are a SaaS company!