Thanks for your question and welcome to the community.
You can still be on a budget, and still safely install the T5 fixtures without issue. I mention this because extension cords aren't designed to be installed as you wish them to be used in your case.
And since these fixtures will be placed inside a high humidity greenhouse, it's a better idea to use higher quality wiring than just cutting corners via an extension cord. In fact, extension cords aren't meant for permanent usage and can become brittle and dry rot over time due to UV exposure.
I would recommend (and it's not that expensive vs an extension cord) to use either exterior grade UF Romex cable or to place THHN wire in 1/2" PVC conduit to connect everything together.
If you do those options, they can give you a better solution for your fixtures. Cheaper doesn't always mean better in regards to everyone's safety.
Let us now if you have any further questions.