From the start we already have a problem. If you have a 13 Dexterity, you don't have the Two-Weapon Fighting Feat as it requires a Dex 15 so it'll take a while to build up to that (probably level 6 minimum to get the appropriate stat buff and have the feat open). That's even without taking Sacred Vow and Vow of Poverty. Without knowing the level and race of the character, this is going to vary wildly, especially with Vow of Poverty.
That said, let's work from the top. 3.5PHB Grappling section on page 156.
Starting a Grapple
If you get multiple attacks, you can attempt to start a grapple multiple times (at successively lower base attack bonus)
So you may initiate a grapple with Flurry's extra attacks
Step 1: Attack of Opportunity
You mentioned going "all in for grapple", so either you've taken the feat or are level 2+. Let's assume either are the case, because even if we assume the level 6 needed for Two Weapon Fighting, your unarmed defense is ~22 with Vow of Poverty which gives you some reasonable chances of not stopping before we begin. Plus a given character can only make an AoO against you specifically once/round, so the next attempt if you have more left won't be subject.
Step 2: Grab
You make a melee touch attack to grab the target
Using a touch attack will help your odds of making the grab in a super majority of cases.
Step 3: Hold
Make an opposed grapple check as a free action. If you succeed, you and your target are now grappling, and you deal damage to the target as if an with an unarmed strike. If you lose, you fail to start the grapple. You automatically lose an attempt to hold if the target is two or more size categories larger than you are. Once again, your build by this point is going to make things extremely varied. For example, if you took "Touch of Golden Ice" as a bonus feat from your Vow and the target is evil.
This is where your success curve will fall. You essentially need to get a second attack off to get you're first hit's worth of damage, and this becomes a wildly floating target. The target might have a great Escape artist check (if their touch AC was high), or they likely have a better grapple bonus than their touch AC modifier; either gets the benefit of a d20 instead of assuming a 10 (55% chance 1d20 >= 10). We're going to wash out size categories for now, but that's +/-4 per difference. On the plus side, at higher levels you get the highest BAB in this check (but so do they)
If you fail the roll, we're back to start like you missed at step 2, so hey - you have extra attacks to fall back on.
Step 4: Maintain Grapple
To Maintain the grapple for later rounds you must move into the target's space. This movement is free and doesn't count as part of your movement in the round. Moving, as normal, provokes attacks of opportunity from threatening opponents but not your target. If you can't move into your target's space you can't maintain the grapple and must immediately let go of the target.
If You're Grappling
You get an action per attack with BAB, and Flurry specifically states it affects the BAB.
- Activate a magic item (Vow of Poverty says you own none)
- Attack Your Opponent: You attack at -4, and do not get the extra attack from TWF ("You can't attack with two weapons while grappling, even if both are light weapons"). They've lost their Dex bonus to AC if any which may offset the penalty
- Cast a Spell / Retrieve Spell Components (You have no spell slots)
- Damage your opponent: Make opposed grapple check to deal unarmed damage.
- Draw a Light Weapon
- Escape from grapple with opposed Escape Artist or Grapple Check against all grapplers opposing you and move to adjacent space (unopposed if they're unconscious/dead)
- Move half speed, bringing whole group with opposed grapple check as a standard action (so only thing with grapple this round)
- Pin opponent with an opposed grapple (which does more for people who aren't you) for one round
- Use Opponent's Weapon is honestly the best thing that can happen in a grapple when you have a Vow of Poverty as long as it's light. You're not taking ownership of the weapon to use it, and it's likely better than the +1 magic weapon that being unarmed effectively rates with the Vow.
Other than the initial "Grab", all ACs for your target include their armor class, and unless they're using Escape Artist to escape a grapple their armor check penalty doesn't come into play (p122). The characters you'll get the most benefit out of grappling are faster ones that you need to nullify Dex and/or movement with, or spell casters so you limit their choices and force concentration checks. Otherwise you're getting a bigger penalty on your attack rolls and essentially drawing a lot of aggro. You're either taking huge penalties to attack, or allowing them a greater chance to resist your attack where they use their highest BAB to resist versus your current one to strike.
All in all I would rate a purely grapple focused monk in D&D3.5e as highly inefficient but it offers some nice options in extremely specific circumstances
I would be more specific but there's a lot of missing information with the build given.