I know that ora pro me means "pray for me", but how would I express my request politely, such as in the English equivalent "Please pray for me" ?
In Latin there is no equivalent for please, you use some form of I ask, instead. Aparently, having a specific word for please dates back just to the Renaissance, and in many languages it comes from more elaborate formulas like if it pleases you, if you are so kind.
I'd offer two possible variants:
Ora, quaeso, pro me (or with a different word order: ora pro me, quaeso, or even with the subjunctive, quaeso ut ores pro me)
The verb quaeso is a favorite in Liturgy, although it is usually in plural (quaesumus). Liturgy is the prayer of the Church as a whole, and hence it is natural for the Celebrant to ask in plural in the name of the assembly. It means to seek to obtain, but also to beg, pray. It is the choice in many prominent prayers, such as the Angelus,
Gratiam Tuam, quæsumus, Domine, mentibus nostris infunde...,/ Pour forth, we beseech Thee, O Lord, Thy grace into our hearts
and the Mass itself (e.g. during the Embolism, after the Lord's prayer),
Libera nos, quæsumus, Domine, ab omnibus malis.../ Deliver us, Lord, we pray, from every evil...
A second option is,
Peto ut pro me ores/I ask that you pray from me.
The verb peto (in singular) was the choice of Saint Thomas Aquinas (XIII century) for his Eucharistic hymn Adoro te devote:
peto quod petivit latro poenitens/ I ask for what the penitent thief asked.
This verb is also used several times in the Gospels, the third time being as early as Mt 7:7:
Petite et dabitur vobis/ ask and it will be given to you
Four more examples of the verb follow in the next four verses.
Update: Draconis notes two important things: that the imperative in Latin is already more polite than in modern languages, and that ut + (subjunctive) is not the most frequent construction in this context.
Peto/quaeso + (imperative) is perfectly valid, adding politeness to the imperative. In singular it sounds a little strange to me, but maybe just because I'm more used to the plural form.
Peto/quaeso (or command verbs) + ut (or its negative form, ne) + (subjunctive) is called an ut clause of indirect command. As infrequent as it is, it remains well attested in both classical and ecclesiastical sources. In order not to make this answer too long, I cite one from the Vulgate,
praecepit illis, ne cui, quae vidissent, narrarent/ He charged them to tell no one what they had seen (Mc 9:9)
and one from the Missal:
Ipsis, Dómine, et ómnibus in Christo quiescéntibus, locum refrigérii, lucis et pacis, ut indúlgeas, deprecamur/ Grant them, O Lord, we pray, and all who sleep in Christ, a place of refreshment, light and peace. (Liturgy of the Eucharist, memento of the dead).
Other possible verbs meaning to ask for something are: rogo and precor (both are already conjugated in 1st person singular, present tense, i.e. they already mean I ask). You could use oro as well, but then you'd be repeating it, since ora is a form of the same verb.
Update: as noted by dbmag9, it is worth to remember, either for you or for future readers, that ora pro me means pray for me as directed to a single person: sencond person singular (thou) and plural (you all, you guys) in Latin are different and you are usually forced to distinguish. If you are asking more than one person to pray for you, you should say orate, quaeso, pro me, or peto ut pro me oretis, or another variant using orate instead of ora or oretis instead of ores.
The simple imperative in Latin is significantly more polite than it is in English. It's even used when making requests to God in Ecclesiastical Latin: pie Iēsū domine, dōnā eīs requiem sempiternam "good lord Jesus, give them eternal rest".
However, if you want to avoid the imperative, there are a few different ways.
One is to use a subjunctive instead:
Ōrēs prō mē
May you pray for me
Another is to add a verb to indicate the request:
Precor tē ōrāre prō mē
I beseech you to pray for me
I'm not sure which would be more fitting for an Ecclesiastical style, though my guess would be the second: the bits of Ecclesiastical Latin I've read tend to prefer extra verbs instead of subjunctives.