We wish to perform backup in SQL Server in 2 different ways in parallel. Now currently we have a 3rd party tool that is performing the Backups, Full and transaction log, now the business wants to perform also backups on disk via local SQL Server jobs. The question is that would be possible, the LSNs chain will be affected as I know, and it won't be possible to perform valid restores, is that correct? Even If we will perform Full backup and Transactional backup on backup server and also full backup and transaction backup to be keep on local disk via SQL jobs. Would it be possible?
There is no way to take additional transaction log backups without affecting the LSN chain. Even taking COPY_ONLY transaction log backups would be completely pointless, as the log chain of the COPY_ONLY backups would be incomplete. You can read more on that here.
You could take additional FULL backups to a local disk without affecting the log chain (if you don't take DIFF backups, the COPY_ONLY flag is not needed, as it only affects the differential base).
If the third-party tool is unreliable in some way, you could back up to a local or network disk and have the backup tool take copies of the backup files instead of taking database backups.
However, if you have issues with that product, I would rather focus on fixing those issues. Many backup tools support a limited subset of the SQL Server backup options and try to treat databases as "files" to better fit in the general purpose backup model they already have. If this is the kind of issues you're hitting, you could also consider dropping the tool altogether and use it to backup your backup files. The main advantage of such tools is centralized management and scheduling for all backups across all instances, which is something you would lose if you switch to SQL Server Agent controlled schedules.