"because if it is, I really am in trouble." <-- haha!
The solution depends on what software your client wants to use.
You can import that illustrator as an image into Indesign without having to re-do the whole layout.
If your client uses Indesign, it's always possible to simply add text frames over the fields you did in Illustrator and prepare some basic texts frames for the contact details, etc.
Your client will be able to print directly from Indesign and/or export a PDF.
Acrobat Pro PDF
You can also export your Illustrator file as a PDF, and add fields that can be typed in with Acrobat Pro.
That method is very reliable as even governmental offices use it for
passports, taxes or residency forms.
There's some nice functionalities to create forms in the Pro version. The fields can be transparent; you can keep your PDF from Illustrator as a background and add the text fields on top of that PDF by simply tracing rectangles (the form field ones) where the editable texts should be. You can lock these fields so they don't move, and if you feel like it, it's possible to add some scripts as well, a bit like Excel does.
But your client might need the Pro version as well, I don't think he will be able to save the PDF WITH the filled fields with a Acrobat Reader version unless Adobe changed this!
Your client will be able to print directly from Acrobat and/or save the PDF with the form filled to their own clients.
Adobe Acrobat DC (Document Cloud)
Word doc / Powerpoint
Exporting to Word doesn't do a very nice job indeed. But it's always possible to export your Illustrator as a PNG or an image that you can place as a background in Word. And then add some text fields over that as well. I don't like Word for this because things often move, and you probably had the same issue.
But your client will be able to print from Word and/or export the file to PDF or simply send as a Word file (not recommended).
Powerpoint has the advantage to be easy to use for the typical office client but it's like using Photoshop for invoices at some point. You can import your file in it and create a template but I think that's about it.
The paper option
Sometimes, you can simply prepare a background layout as any other invoice and print them. Then create a digital template with Word, Indesign or whatever the client prefers and add text frames where the details need to be edited.
That method is very 1990 but it works too.
Then print these new data on the sheets that already have your basic invoice printed! The same way as you would do with a letterhead.
Obviously, this can be annoying to set up, often people forget to remove or put the right paper for this in the printer, but it's another option that was used for a long time in every office and is still used today.
Unfortunately, unless the client uses a software that permits a "no print" option on some elements (the background), he won't be able to send a PDF with the background layout + the data. Unless he scans it, which is a bit counterproductive.
If you client expects a totally free solution, there's OpenOffice (similar to Microsoft Office) but it's openSource and a bit buggy.
Otherwise he needs to pay for an Acrobat Pro ($14/mt) or Acrobat PDF pack ($89/year) or Microsoft Office ($6/mt). It's still a bit cheaper than most invoicing systems ($15 to 100+) but nothing is free. At this point it's a matter of what's most practical and what does the job. The PDF/Word is more flexible for design, scripts and fields and the invoicing system is more on "auto-pilot" but not as versatile on these points.
Don't forget the fonts!
In all cases, don't forget that if you prepare an editable template for your client, you need to make sure your client will be able to use the same exact fonts as yours!
Either you find a free online font the client can install, use Typekit if he has an Adobe subscription or use some basic safe fonts for the fields that he needs to update (eg. georgia, arial, verdana, etc.)