To some extent what you are asking is how best to build a compelling "business case" for the things you are after. In order to build a compelling case it can be useful to "seek first to understand, then be understood."
In building a case, the two main things to stress are :
- we are at risk and this removes the risk
- this is an investment, and will deliver rewards
In my experience most organisations respond more rapidly to risk (fear) than reward (greed), so when building the case this can be a better argument. More importantly, perhaps, people tend to be very skeptical of efficiency improvements, unless they either allow them to reduce staff headcount or produce a compelling improvement (usually a x10 factor, or 5%-10%+ of your total working time in a week)
So - I would suggest where possible you try and link the improvement you are after to risk reduction, unless you can make a compelling case for cost/time savings.
Examples might be:
- reducing risk of employee injury (downtime, legal implications)
- reducing risk of employee flight/turnover (poor conditions, equipment)
- reducing risk of poor client/investor perception (outmoded hardware, scruffy offices)
The second thing to consider is where the money comes from.
In most organisations there are two parts to this - whether your boss has the "sign off" level to approve these things directly, and whether they have a budget set aside for this kind of thing.
These two issues are important, because if your boss lacks the either the direct budget control or the sign-off, then they will have to go to another contested pot of money elsewhere in the organisation (HR, IT, or their line manager), which will generally be a lot more work (for your boss), and require a much more compelling case.
Consider the situation where your boss has to make the health and safety case to someone else for the chair - a senior manager, or HR perhaps. if they take action in your case, they will have to take action across the whole organisation. The costs are no longer $500, they are $500 for everyone with an older chair.
If this might be the case, then its usually better to raise the issue informally with your manager, and make it about the whole organisation from the outset, not just about you, and to lead in with the key argument and supporting evidence:
"I came across this article that suggests poor ergonomics costs over $$million a year, nationally, in terms of employee downtime and workplace injury cases. Looking around this is a real risk for us - how can we fix it?"
Indicating from the outset that you are aware of the implications for your boss of what you are asking, and feeding them the ammunition they will need to make the case with others makes it much more likely they will respond in a positive way.