No, multiple clients cannot write into an HDFS file at same time.
When one client is given permission by Name node to write data on data node block, the block gets locked till the write operations is completed. If some other client requests to write on the same block of a particular file in data node, it is not permitted to do so. It has to wait till the write lock is revoked on a particular data node. All the requests are in the queue and only one client is allowed to write at a time.