I have the same version of Open Office and, interestingly, I just had a crash while trying to play around with this form. I can replicate the crash if I tab beyond the "end" of the form trying to jump to the next (and non-existent, in this case) fillable item.
The above being noted, the form itself probably has accessibility issues. I have not used "fillable forms" in Open Office but have in MS-Office and, based on what you've supplied, this thing appears to be a form template but not set up that the end user can only enter data in specific form fields. Some of what you've filled in shows up in advance of the text boxes that I presume the developer probably thought you'd be landing in and then typing in. There is only one checkbox, at the bottom of page two, and even when I click on it with the mouse the object itself is not announced (making me believe there's no alternate text that describes its purpose) nor do I get any feedback from NVDA when I check or uncheck directly with the mouse.
I took it upon myself to "go to the source" and this form downloads as an MS-Word document. When opened under MS-Word it behaves as I'd expect - a template with fillable fields and those fields are the only thing you can touch. It's still horrible from an accessibility standpoint because the fields do not have any descriptive text associated with them to tell you what you've landed in and the checkbox is the same. Checkbox is announced but the question would be, "Checkbox for what?"
If I open the MS-Word document under Open Office or Libre Office, even retaining that format and not converting to ODT format, neither behaves at all well with this form. They both go back to treating it as if the whole document should be editable rather than traversing just the form fields and letting you fill them in.
I'm sorry that this is what I have to report. Given "the source" of this form it would be entirely legitimate to complain loudly that the form itself, in MS-Word, is not accessible in its native format. They need to add at least descriptive text, if not that and help text, to the various form fields so that anyone accessing it with Word and a screen reader can have a clue as to what they're expected to be filling in when they land on a given field or are checking to indicate on a checkbox. I doubt you'll get any traction as far as compatibility with OpenOffice or LibreOffice. Organizations don't, and can't, create electronic forms for every conceivable office suite. That being said, if they wanted to be more "universal" about it they should be using fillable PDFs, which are supported under Adobe Reader and can be saved after being filled in.