Answer: While it may take some time for the plant to cover the frame, your project is basically pretty simple. You could grow a shrub next to it and train it to match the frame or you could cover it with a vine.
The evergreen shrub called yew is a good choice for training to match a frame and has been used extensively for topiary. However, since your creature is a four legged one, it will be very difficult to make that shape out of one shrub unless you allow the shrub to work as a plinth (or base support) at first, and then begin training four branches upward, one along each leg. Essentially, this will put your deer on a living pedestal.
Simply tie the selected branches to the deer frame, allowing them to grow to the top of the frame, then trimming them to encourage branching. If you keep the main skeleton of wood inside the frame and trim often enough using the frame as a guide, it will grow dense and look "solid." In the meantime, the frame helps support the plant, so I hope your frame is quite sturdy. (Traditional topiary frames are made of iron.) Be sure to check your ties and loosen them as the branches thicken with age. Eventually they should assume enough strength that it becomes fairly self-supporting. Yews can become rather large plants and ultimately it may become difficult to keep it as small as the frame you have -- often these topiaries are quite a bit larger than life to facilitate training the plants.
If that sounds a bit problematic, you might prefer to try a vine. To keep it outdoors you will need a hardy plant to grow on it and English ivy (Hedera helix), an evergreen, will do the job. A smaller-leafed variety looks best. Plant it at the base of each leg and train it to the frame. Planted in rich moist soil in a sheltered location, the ivy should grow fairly quickly. Pinch out the tips as needed to encourage it to branch and spread.
If you could keep it indoors for the winter and set it outside for only the summer, then the tropical houseplant called creeping fig will cover it very nicely. (I believe this is what is often used in Florida.) You can plant this in pots inset into the frame as well as set each leg into a larger container holding a few plants. Creeping fig grows quickly if it is kept well fertilized and watered, and is misted frequently. Be sure to acclimate it slowly when moving it in and out of doors. (You could also do the same thing with English ivy.)
For both vines, line the frame with a material such as sphagnum moss from the craft shop to help the topiary look better while it is "in progress". When you water the plant, make sure to water this liner material also. This will give the vines more to cling to while they grow, and provide extra humidity and moisture.
Have fun with your project!
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