For sports photography with the Sony a7III, I'd highly recommend checking out these two lenses:

1. Sony FE 70-200mm f/2.8 GM OSS II

• A pro-grade telephoto zoom with a constant f/2.8 aperture
• Super sharp optics and fast autofocus for capturing action
• Built-in image stabilization (OSS) for handheld shooting
• Ideal focal range for sports like football and basketball
• Excellent low-light performance when you need it

2. Sony FE 100-400mm f/4.5-5.6 GM OSS

• Versatile super-telephoto zoom range
• Relatively compact and lightweight for a lens in this class
• Fast, quiet autofocus and responsive handling
• OSS helps when shooting handheld at longer focal lengths
• Well-suited for outdoor sports in good lighting conditions

Between the two, the 70-200mm f/2.8 GM II is the more popular choice for indoor sports and low-light situations where the extra stop of light gathering makes a difference. The 100-400mm is great if you need the extra reach for field sports and want to travel lighter.

Both lenses are pricey, but deliver uncompromising image quality and performance for professional sports photography. They're well-built and fully weather sealed too. Renting is a good option to try them out before buying.

For a more budget-friendly alternative, look at the Sony FE 70-200mm f/4 Macro G OSS II. It's smaller, lighter and less expensive while still delivering excellent sharpness and AF speed. You'll just have to bump up the ISO more often in lower light.

Hope this helps with your decision!

I love the Sony FE 200-600mm lens, it is a flexible zoom perfectly suited for nature, wildlife, and sports applications. Its long reach pairs with a modest maximum aperture range to render to a relatively lightweight and portable design and, as a G series lens, optical quality, focus speed, and attention to handling details are also priorities of this well-rounded super-tele zoom.