You can run your own web-server in house, and develop and host your site on that, but frankly there is little point. You would have all the expense, and the hassle of maintaining the server 24 hours a day, every day and ensuring that you had enough bandwidth to deal with the traffic as your business grows.
A far better solution is to host your site with one of the reputable hosting companies. You'll have a flexible business package that can grow with the business, and your site will be hosted in a state of the art server facility - either on a shared server, or a dedicated one if your traffic warrants it.
You can develop the site on your office computer(s) and upload changes at will, any time you like. Most small to medium sized businesses do it that way.
I have run sites with this company for a number of years, and have found them to be excellent - plenty of options, and a very high level of service.
Companies like this understand the needs of business users, and your hosting package will come with a comprehensive range of sophisticated tools and facilities. E-commerce integration is a given - pretty well every business runs an online shop these days.