I expect it to be very hard to find a good low-aperture telephoto zoom lens under 200$. The only telephoto zoom lens under 200$ that I am aware of is the Canon EF 75-300mm f/4-5.6, at 199$.
Ideally when looking for a telephoto lens, you want to look for one with image stabilisation. As a side note, this changes names based on which company makes the lens… Canon – Image Stabilization, (IS) Nikon – Vibration Reduction (VR), Sigma – Optical Stabilization (OS), Tamron – Vibration Compensation (VC), Leica – MegaOIS, Pentax – Shake Reduction (SR). Ideally you want f/4 or f/2.8, depending on the kind of work you do. But, if you want an f/2.8 telephoto zoom lens with IS you are looking north of 1000$ at best.
If you don’t mind having a prime lens, then you might actually get what you are wanting by getting a cheap third-party lens, such as the Meike MK-85mm f/1.8. This is a cheap lens, and has its issues, but you can’t really do better in terms of fast lenses (large-aperature) for the price.
If you want to save a bit you could also look at used lenses, ideally from a reputable company. At the time of writing, B&H has a used Canon EF-S 55-250mm f/4-5.6 IS II for 90$ and a 55-200mm f/4.5-5.6 USM II for 110$. The IS version has Image Stabilisation and the USM version has the built-in focus motor. Even used, you have to go up to 560$ before getting to a Canon EF 70-200mm f/4L USM (a great lens, though without IS) or the Tamron 70-210mm f/4 Di VC USD, which has VC.
Sorry for the rambling… I hope this was more helpful than discouraging!