When faced with a poor chair I have seen several solutions to the problem:
steal a chair from another office or the conference room. This is harder to do if the ones in the conference room are a different style than the ones used in offices. You don't want to get caught doing this so don't do it while everybody is watching.
ask the office manager or your supervisor. They might just get one from an empty office for you or they may have to get one from storage or even order one. You could say "Is there any way I could get a different chair, this one is ... and causes me a lot of back pain?"
The laptop issue is serious for the company. Your personal machine might not be automatically backing up key documents according to company policy. The machine might not be running virus protection, or it might be out of date. Some companies even require full disk encryption to protect company and customer data. There are also licensing issues for the software on your machine, and the fact that they might have to install software they paid thousands of dollars for onto your machine. Which you could keep by walking out the door.
You have to ask your manager about getting a company machine. List the issues discussed above and then ask for a timeline for resolving the issue. You will also need to talk about the type of machine need to do the job. In a small company without a big bureaucracy they should be able to submit the order in a day or two and get the machine delivered not long after that.
The one issue that also needs to be understood is your contract. Are you and employee or a contractor? In some cases they expect the contractor to supply their own tools, in other cases they provide the tools.