This depends a lot on your company. As you say, it's not technically advanced, and as such solutions that depend on technical changes may be somewhat hard to implement.
Stable storage format
I would start in your own area and devise a scheme in which you organize the collected data. As a software developer, I'm always inclined to develop an automatic solution, but you're probably better off with a manual process that you follow reliably. This may be as simple as having Excel files organized in folders by reporting period. The key is that you don't change the format and process too often, so you get familiar with it, and maintaining the data becomes fluent over time.
For the data that currently needs to be collected using e-mail, I would try to meet with the people who have access to that data to find ways of automatically extracting it. If it's in databases, something like a scheduled script to perform some SQL and send you the results might be enough. Setting up the first of such scripts will likely be a bit of work and will require the assistance of knowledgeable colleagues. The work will become easier once you have done it for one or two data sources, as you then know whom to ask and how to best format the data.
More automation - maybe, maybe not
You might be tempted to add more automation so that data collected from the different sources is automatically consolidated. I would not plan for that until you have sufficient experience with manually performing the process, so you really know what works and should be automated. However, you should also consider that part of data collection is quality control, and it might be better to keep the manual consolidation step as you are able to see data inconsistencies early. Putting such checks into code is possible but harder than you might think, and probably not worth it, depending on the volume of data.