There are two aspects to this question.
Data migration
Your first step will be to "Define RDBMS schema" but how far are you going to go with it? Spreadsheets are notoriously un-normalized and so have lots of duplication. You say in your other question that "Data is loosely structured, and there are no explicit constraints." If you want to transform that into a rigourously-defined schema (at least 3NF) then you are going to have to do some cleansing. SQL is the best tool for data manipulation.
I suggest you build two staging tables, one for each worksheet. Define the columns as loosely as possible (big strings basically) so that it is easy to load the spreadsheets' data. Once you have the data loaded into the staging tables you can run queries to assess the data quality:
- how many duplicate primary keys?
- how many different data formats?
- what are the look-up codes?
- do all the rows in the second worksheet have parent records in the first?
- how consistent are code formats, data types, etc?
- and so on.
These investigations will give you a good basis for writing the SQL with which you can populate your actual schema.
Or it might be that the data is so hopeless that you decide to stick with just the two tables. I think that is an unlikely outcome (most applications have some underlying structure, we just have to dig deep enough).
Data Loading
Your best bet is to export the spreadsheets to CSV format. Excel has a wizard to do this. Use it (rather than doing Save As...
). If the spreadsheets contain any free text at all the chances are you will have sentences which contain commas, so make sure you choose a really safe separator, such as ^^~
Most RDBMS tools have a facility to import data from CSV files. Postgresql and Mysql are the obvious options for an NGO (I presume cost is a consideration) but both SQL Server and Oracle come in free (if restricted) Express editions. SQL Server obviously has the best integration with Excel. Oracle has a nifty feature called external tables which allow us to define a table where the data is held in a CSV file, removing the need for staging tables.
One other thing to consider is Google App Engine. This uses Big Table rather than an RDBMS but that might be more suited to your loosely-structured data. I suggest it because you mentioned Google Docs as an alternative solution. GAE is an attractive option because it is free (more or less, they start charging if usage exceeds some very generous thresholds) and it would solve the app sharing issue with those other NGOs. Obviously your organisation may have some qualms about Google hosting their data. It depends on what field they are operating in, and the sensitivity of the information.