It is possible that living in rented accommodation could stop you from having a puppy. Puppies require a lot of care and if you live alone then you will be leaving it on its own during the day if you have a job, and things could get very messy if you do not have the time to look after it properly. Leaving the house in a mess would obviously not go down well with your landlord.
The best thing to do would be to discuss the matter with your landlord and see whether or not they would be okay with you having a puppy.
Some landlords may be okay with you having pets around the house, but the majority of them prefer you not to have them, as pets can be messy and if you aren't around to train it or look after it properly then it will surely make a mess of the house.