From its early days, the nascent field of computational social science has been dealing with questions of reproducibility and replicability. While some of these questions are similar to other fields within the behavioral and social sciences, several challenges are unique to or at least more pronounced for computational social science. The main reason for this is the type of data that computational social scientists typically work with, most of which belong to the category of so-called digital trace data. These data are generated by users of digital technology, come from a variety of sources and in many different formats, and are most often controlled by the companies operating the platforms and services. Hence, many researchers have to rely on data access methods offered by private companies, such as Application Programming Interfaces (APIs). These, however, can be changed or closed off altogether, thus, impacting the replicability of research that utilizes them. Alternative models of data access, such as data donation from users, have been proposed and tested by researchers, but they are much more costly and not trivial to implement. Also, these alternatives are apt to severe sampling biases. To address these challenges, it may help to pool resources by establishing something akin to the distributed laboratory network Psychological Science Accelerator for the collection of digital trace data. Another challenge that computational social science needs to address is that of sharing digital trace data. The volume, format, and sensitivity of the data as well as potential requirements from the companies that control them place restrictions on data sharing. Similar to data access, this requires novel (technical and organizational) solutions, such as non-consumptive data use or secure remote access. This presentation discusses these challenges and introduces potential solutions, such as software archives, detailed documentation of materials and methods, and controlled data access.