Store tends to be a slightly more american usage, the typical spanish small town "supermercado" would most likely be referred to as a shop in British/Irish English. Supermarket is commonly used here (UK and Ireland), but they are more like the French hypermarkets, big out-of-town stores (como "El Corte Inglés", "Carrefour"). But repeating the word shop "I have shopped at this shop for 8 years" sounds a bit odd, so my translation might be "I have been shopping here for 8 years".

I have shopped in this supermarket for eight years. But, you wrote I bought from this supermarket since 8 years. The first I wrote is correct. I have shopped (here the supermarket) for eight years.

El primer ejemplo de Michael me parece un poquito raro con "since". La traduccion mas simple (y mas directa) en mi opinion es... "I have shopped in this store for 8 years." ("supermarket" existe en ingles, pero es mas comun decir "store"...a menos que quieras especificar el tipo de mercado/tienda.")

We would probably change it to present perfect and say "I have shopped in this supermarket since 8 years ago" or "I have been a customer here for 8 years".