What version of the Avamar clients are you running? Was it ever working before? What does the SharePoint backup log on the SQL server say? Are the VSS writers related to SQL and SharePoint in a stable state on both servers?

Version is 6.1. On the SharePoint server there is Avamar windows and SharePointPlugin. On the SQL server there is Avamar windows and SQL plugin (because the are more instances and databases, different type of backup). Based on your question do I need to install SharePoint plugin on the SQL server? VSS writers are OK.

Here are errors:

2015-07-22 03:05:37 avmossvss Error <0000>: The back-end client [SQL] will be dropped later in the processs. Please examine its activity logs for more information. (Log #4)
2015-07-22 03:05:40 avmossvss Error <0000>: The back-end client [SQL] will be dropped later in the processs. Please examine its activity logs for more information. (Log #4)
2015-07-22 03:10:56 avmossvss Error <13968>: The back-end client [SQL] has failed to backup, so its components will not be displayed for restore. (Log #4)

and more:

2015-07-22 03:00:37 avmossvss Info <13994>: Preparing a manager for the federated backup. 
2015-07-22 03:00:37 avmossvss Info <12962>: Master plugin runs on: '127.0.0.1' 
2015-07-22 03:00:37 avmossvss Info <13995>: Sending sub-workorders to the back-end clients. 
2015-07-22 03:00:37 avmossvss Info <13962>: Number of back_end_clients: [1] 
2015-07-22 03:00:37 avmossvss Info <12978>: Forwarding [snapup] request to: 172.16.10.46 
2015-07-22 03:00:37 avmossvss Info <13967>: Back-End Client: SQL,172.16.10.46, Status: SENT_SUBWORKORDER 
2015-07-22 03:00:37 avmossvss Info <13088>: Waiting for back-end clients to start sub-workorders ... 
2015-07-22 03:05:37 avmossvss Info <13089>: Initiating VSS Snapshot creation on back-end clients
2015-07-22 03:05:37 avmossvss Error <0000>: The back-end client [SQL] will be dropped later in the processs. Please examine its activity logs for more information.

2015-07-22 03:10:56 avmossvss Info <8393>: Generating snapview... 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\User Profile Service Application_SyncDB_fe6a520626764393aa0a633587f2ea50 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\3445ee04-0f4e-4e5e-a54f-b7ac19304e81\3afc0499-1108-4864-a99e-c065a433f170\Search_Service_Application_CrawlStoreDB_00f6da6ac8fb4d5bb0dbdc563f42ecfb is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\Secure_Store_Service_DB_3cf70fee8d4148e3ac00c2aa5be618e9 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\Managed Metadata Service_e8e7955fb51c4015a97b53a9e600442f is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\Bdc_Service_DB_54fc9bb88a234982b1ec66d5dc807624 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\3445ee04-0f4e-4e5e-a54f-b7ac19304e81\066b30cd-1125-49b4-9eaf-d579b1a70e3b\Search_Service_Application_DB_b7ede9ef989c4578aee122610d2cfd74 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\WordAutomationServices_f4ec93fef11c4d4583d523707e98653a is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\3445ee04-0f4e-4e5e-a54f-b7ac19304e81\604d813b-435a-4322-a02f-5b786e2a4a1e\Search_Service_Application_AnalyticsReportingStoreDB_690169af366c455c8fa0f54386949e31 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\StateService_70c4ab661b934a98ba08cff4ca94801f is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\User Profile Service Application_SocialDB_b12e6e52ded748d1838b486bfa684984 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\SharePoint_AdminContent_677ad23b-1e53-41d8-b60a-9e5d4a19d757 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\3445ee04-0f4e-4e5e-a54f-b7ac19304e81\4ef6edca-1f1e-4503-991e-6def0c6f52df\Search_Service_Application_LinksStoreDB_91107438b458454ca7765913758b72ef is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\PerformancePoint Service Application_cbc3847d965d4bffa660e278dc2e5fb9 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\AppMng_Service_DB_3e24c16fb9604339bd04e29ec453f0c4 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\WSS_Content is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQL\TranslationService_defdf37c25834ae789e4e752e6f32983 is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Warning <0000>: Component {da452614-4858-5e53-a512-38aab25c61ad}:\SQLUser Profile Service Application_ProfileDB_0f26bbe6c49f46999fb4571bcf3b8f8f is not successfully backed up and will not be displayed for restore. 
2015-07-22 03:10:56 avmossvss Error <13968>: The back-end client [SQL] has failed to backup, so its components will not be displayed for restore.

Thank you for your answer ill try that later