The Prophet ﷺ performed tawaf around the House on his camel, touching the Black Stone with his crooked staff, and he also went between Safa and Marwa in the same manner.Yazid once said, "On his mount, he would touch the Black Stone."
حدثنا يزيد أخبرنا مسعر بن كدام عن عمرو بن مرة عن سالم بن أبي الجعد عن أخيه
عن النبي ﷺ أنه طاف بالبيت على ناقته يستلم الحجر بمحجنه وبين الصفا والمروة وقال يزيد مرة على راحلته يستلم الحجر
Its isnad is sahih. The Messenger of Allah's ﷺ performing tawaf while riding his mount is established in hadiths from Ibn Abbas and others. See 1814, and see Muntaqa 2562–2566.