If you expect fair growth or change in your demands within 5 years, it makes sense to look into the well known parties that can rent you infrastructure: Amazon, Microsoft and maybe Google. The choice would depend on your specific demands. Buying your hardware could be cheaper, but that supposes you use your hardware fairly optimally and depends on your needs. That means correctly projecting usage and then using your hardware consistently. Considering it is unlikely that you will use most of your hardware from the start and not grow out of it after that, it seems to make sense to rent infrastructure. You can scale up and down according to your wishes. When things settle down and growth is more predictable and less volatile, you can start looking into buying your own iron. You could always use a hybrid solution, using rented infrastructure as a backup for temporary spikes or specific types of usage.
Of course, choosing cloud services has serious consequences. You will need to put a lot of effort in making sure your customer's data is secure and stored correctly, which also means making sure that it stays in the correct countries and does not get backed up to the wrong countries. Obviously, that is also the case if you have your own hardware. You, as a company, are still responsible for where the data might end up.
Also plan for problems. Even the biggest companies have outage issues at times. If the data logging is mission critical for your customers, you will need to have some kind of failover contingency plan. Be sure to be contractually very clear about what they can and cannot expect, but pointing at Amazon because a client lost irreplaceable data is not going to make you friends.
Of course, whatever you do, make sure you have competent people creating and maintaining your infrastructure, virtual or physical. It does not take too much for a major liability to arise if someone is fudging things. Make sure security is not an afterthought. Proper design and maintenance with attention for security from the start is key.