How can I speed up backups for my Enterprise Vault backup Netbackup 7.0.1- EV 9.0.1

There is nothing much you can do on Ev part , only thing which I can think of is

1. Enable collections on you vault store. Do you have collections enabled on your vault stores? this will compress the files in your vault store in a .cab file. Thus the time to backup would reduce.

2. Create Full -incremental backup stratergy so you just backup the changes on daily bases and over the weekend you run a full backup .

On Netbackup side, if you thorught put is less compare to the other backups you need yo check if any network related issue between the media server and the Ev server>

We are not using the EV agent, and I had to split the backup into multiple streams in order to back it up successfully.  This was even after turning on collections.  Example:

NEW_STREAM
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2009\
NEW_STREAM
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2010\
NEW_STREAM
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2011\01*
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2011\02*
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2011\03*
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2011\04*
H:\Enterprise Vault Stores\MSPMBAVS01 Ptn1\2011\05*

In addition, I had to create multiple policies with different schedules for full backups so that we don't hit all of the directories on the same night for fulls. 

if flatfile is what your motto is, try flash backup and check if it helps

Would recommend to go for EV enabled agent based backups. Though EV backup are likely to be slow in terms of performance, as there are more chunks of files which gets indexed during backup hence lengthier backup windows.

Collections are great for BACKUP but affects to endusers from EV perspective unless EV is enough strong to handle decompression load of simultaneous requests. as in collection all data would be compressed in .cab RESULTING SINGLE FILE for backup (faster backups compared to multiple files), but affects EV performance. Choice is yours

Another thing you can consider (as well as Collections) is to close off some of the 'older' partitions, and implement a process to close/create-new-open-partition regularly, eg every quarter? every 6 months? every year?