I was wondering if it's possible to avoid buying a $400 48V 15Ah battery with an output of 54.6V and 2A, and instead make one by joining multiple 3.7V 5Ah batteries in series and or in parallel?
Yes, this is definitely possible, but lithium ion batteries used in series arrangements require extra provisions for safety, including cell matching, where cells are tested in large quantity and binned for their properties, and only used with other brand new cells with matching properties. This can allow for the simplest charger arrangements. Individual cell monitoring and galvanic isolation can be used as well.
Is a conversion like this even possible with the batteries I have? If not, what batteries would I need?
If you have brand new cells you can test them to see how well matched they are. If you don't have a sufficient quantity you may wish to order the full number to avoid matching problems with your existing ones, or look for "13s3p batteries" on the google to find pre matched and connected sets for diy purposes. If you do decide to build your own, be sure to use a charger and battery protection that suit the purpose, and don't neglect the option of refurbishing an existing battery pack, as the value of a well engineered, safe, lightweight plastic casing can be substantial.