(1) Background: In response to the global COVID-19 pandemic, governmental measures have been undertaken. The impact of the crisis on the healthcare of patients with cancer is largely unexplored. This multicenter cohort study aimed to investigate a potential screening delay and its consequences in patients with oral cancer (OC) during the pandemic. (2) Material and Methods: Data of patients who were first diagnosed with OC during different periods were collected, especially in terms of OC incidence, tumor stage/entity and time to intervention. The periods lockdown (LD) (13 March–16 June 2020), post-lockdown (PLD) (17 June–1 November 2020), and the corresponding equivalents in 2018/19 were differentiated and compared. (3) Results: There was no obvious trend towards a higher incidence of OC or higher tumor stages, whereas a trend towards a shorter time to intervention during the LD2020 could be observed. Subgroup analyses revealed an increased incidence in OC within the PLD2020 in Mainz, which might be explained by the partial closure of dental practices in this federal state during LD. (4) Conclusions: While there was no overall higher incidence of OC, we found closure of practices during LD to possibly delay cancer diagnosis. Therefore, measures must be taken to identify patients at risk and to ensure basic healthcare, especially in the context of dental screening measures.