JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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I have checked your forms and I can see you're not currently using a payment field, but if you're planning to integrate your form with a payment method and generate invoices, then you may consider to use FreshBooks, here is the link to the integration: https://apps.jotform.com/app/freshbooks_integration
Here's another link that may help you: Create-Invoices-from-Forms-Using-Freshbooks-Integration
This does not need you to upgrade, but do note that as a free account you only are allowed to receive 10 payments.
But if you're looking for invoices for your account, then you need to upgrade, then you will find the invoices on your account, here is a guide that will help you to check your invoices: How-to-Check-Invoices
I believe you misunderstood me. If I want to upgrade our account because we need more form submitted than the free account allows for, can JotForm invoice my company or do we need to pay online using a credit card? For my company's purpose, getting a bill/invoice is easier than putting it on a credit card and having to expense it.
Yes, I misunderstood your question, apologies as I thought you were referring to invoices on your account or for your customers.
Now, regarding to pay your subscription, you can do it with a Credit Card, you may also use a PayPal account, with these methods you can pay a monthly, yearly and bi-yearly subscription.
You may also purchase a subscription via Check/Purchase Order/Wire Transfer, but as these methods required manual process you can only purchase yearly or bi-yearly subscriptions, monthly subscriptions are not allowed with these payment methods.
Hope this information helps.