In such a scenario, how much would you expect the total costs would be and what service would you use?
There are preset and custom plans available. There is a free tier that may work if what you're scraping is on their whitelist; all paid accounts are unrestricted that way.
If you're not doing any computationally intensive processing, and it sounds like you're not, the biggest factor in cost will probably be how large your database is going to be.
It did everything I needed for $5USD/month.