ALL THESE YEARS Moscow has been on the outside of peacemaking in the Arab-Israeli dispute. Now it prepares to be on the inside. It is putting new steam behind its old favor for an international conference -- a format that would remove Mideast peacemaking from exclusive American auspices. To assist in this venture it is broadening contacts with Israel. Two decades after breaking off relations with Jerusalem, it has opened a consular mission in Israel. The new office gives the Kremlin, which has long recognized the PLO and pursued close ties with Arab states, a would-be broker's minimum equipment of access to both sides.

Up to now the Kremlin has countenanced a Western lead in Arab-Israeli peace diplomacy and has used arms sales and political support for hard-line Arab regimes to play a spoiler's role itself. Nor has it forsaken that line. The Soviets took a part in the recent rescue of the PLO's ''rejectionist'' elements from isolation within the Palestinian movement. They arm Syria at a level responding to Syrian ambitions for ''strategic parity'' with Israel.

The Soviets are promoting an international conference as a vehicle for their own diplomatic penetration. But they seem uncertain on how to make that vehicle go. They show an understanding for American insistence that a conference not be empowered either to veto an agreement negotiated directly by the parties or to impose a settlement of its own. But then they proceed to fuzzy talk to the effect that a conference ought to be ''authoritative.''

Israelis are torn between wanting to exclude the reality of Soviet hostility and wanting to explore the potential of Soviet moderation -- moderation that presumably would be reflected in Soviet Jewish emigration as well as in regional diplomacy. At present, they are deadlocked on the conference proposal. The prime minister's party, fearing the call for compromise that would arise at a conference, says no. The foreign minister's party, readier for such a call, says yes. Deadlock means no, but the Soviets now can start pleading their case in person.

The initiative in Arab-Israeli diplomacy, when the Reagan administration came to power, was largely in American hands. Now it's moving in Moscow's direction. This could be the administration's principal legacy in the region.