Hi. Welcome to Just Answer.
I have seen many units like you describe, both manufactured, and stick builts with sub floors.
You could call in a duct cleaning service but rest assured you wont like the cost.
The units that I have cleaned I did with a good powerful shop vacuum and some flexable tubing as an extension (see picture). The last two I hired a college kid to do for me. Knees dont crawl very well anymore.
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the bill is going to the previous owner as this was supposed to be taken care of prior to my assuming ownership.
given that, would a duct cleaning company be willing to do this? or, would they not want to be dragging their equipment around in filth?
Jennifer, I really coudn't say. It would be up to the company.
I can tell you that most duct cleaning companies that do HVAC ducting have to do systems under houses as well as inside them. I just dont know if there are any local companies that do duct cleaning in your area.
Please provide your zip code and I will see who is available in your area.