German employment law does not generally require employment contracts to have a specific form. An oral agreement is enough, as is the simple commencement of work or the exchange of emails.
But beware: some agreements must comply with the statutory written form to be effective. This means wet-ink signatures of employer and employee. Notably, this applies to fixed-term employment contracts (befristete Arbeitsverträge), post-contractual non-competes (Nachvertragliche Wettbewerbsverbote) and apprenticeship contracts (Berufsausbildungsverträge).
Except for these exceptional cases, however, the use of DocuSign or other e-signing solutions for employment contracts generally also works. However, at the latest at the start date, the employee must be given a signed record of the essential contractual conditions by the employer. As of August 1, 2022, a violation of this is even punishable by a fine of up to EUR 2,000.00 per violation but would not result in an invalidity of the employment agreement per se.
Such essential contractual conditions of the employment relationship include, among others: